2.1 The affine Weyl group and the associated Hecke category

We let $\Bbbk$ be an algebraically closed field of characteristic $p$ (possibly equal to $0$), and $\mathbf {G}$ be a connected reductive algebraic group over $\Bbbk$. We fix a Borel subgroup $\mathbf {B} \subset \mathbf {G}$ and a maximal torus $\mathbf {T} \subset \mathbf {B}$. The Lie algebras of $\mathbf {G}$, $\mathbf {B}$, $\mathbf {T}$ will be denoted $\mathbf {g}$, $\mathbf {b}$ and $\mathbf {t}$ respectively. We set $\mathbf {X}:=X^{*}(\mathbf {T})$ (respectively, $\mathbf {X}^{\vee } := X_*(\mathbf {T})$), and denote by $\Phi \subset \mathbf {X}$ (respectively, $\Phi ^{\vee } \subset \mathbf {X}^{\vee }$) the root system (respectively, coroot system) of $(\mathbf {G},\mathbf {T})$. The canonical bijection $\Phi \xrightarrow {\sim } \Phi ^{\vee }$ will be denoted $\alpha \mapsto \alpha ^{\vee }$. The choice of $\mathbf {B}$ determines a subset $\Phi ^{+} \subset \Phi$ of positive roots, consisting of the $\mathbf {T}$-weights in $\mathbf {g}/\mathbf {b}$; the corresponding basis of $\Phi$ will be denoted $\Phi ^{\mathrm {s}}$. In this section we will make the following assumptions.

1. (i) $p$ is good for $\mathbf {G}$.

2. (ii) Neither $\mathbf {X}/\mathbb {Z}\Phi$ nor $\mathbf {X}^{\vee }/\mathbb {Z}\Phi ^{\vee }$ has $p$-torsion.

3. (iii) There exists a $\mathbf {G}$-equivariant isomorphism $\mathbf {g} \xrightarrow {\sim } \mathbf {g}^{*}$.

For simplicity we will fix once and for all a $\mathbf {G}$-equivariant isomorphism $\kappa : \mathbf {g} \xrightarrow {\sim } \mathbf {g}^{*}$. By equivariance this also provides an identification of $\mathbf {t}$ and $\mathbf {t}^{*}$ (where $\mathbf {t}^{*}$ is identified with the subspace in $\mathbf {g}^{*}$ consisting of linear forms that vanish on all root subspaces).

Let $\mathbf {W}=N_{\mathbf {G}}(\mathbf {T})/\mathbf {T}$ be the Weyl group of $(\mathbf {G},\mathbf {T})$. The associated affine Weyl group is the semi-direct product

\[ {W_{\mathrm{aff}}} := \mathbf{W} \ltimes \mathbb{Z} \Phi \]

where $\mathbb {Z} \Phi \subset \mathbf {X}$ is the lattice generated by the roots. For $\lambda \in \mathbb {Z}\Phi$ we will denote by $t_\lambda$ the image of $\lambda$ in ${W_{\mathrm {aff}}}$. It is well known that ${W_{\mathrm {aff}}}$ is generated by the subset ${S_{\mathrm {aff}}}$ consisting of the reflections $s_\alpha$ with $\alpha \in \Phi ^{\mathrm {s}}$, together with the products $t_\beta s_\beta$ where $\beta \in \Phi$ is such that $\beta ^{\vee }$ is a maximal coroot. Moreover, the pair $({W_{\mathrm {aff}}},{S_{\mathrm {aff}}})$ is a Coxeter system; see [Reference JantzenJan03, §II.6.3]. We will sometimes need to ‘enlarge’ this group by considering translations by all elements of $\mathbf {X}$. Namely, the extended affine Weyl group is the semi-direct product

\[ {W_{\mathrm{ext}}} := \mathbf{W} \ltimes \mathbf{X}. \]

Then ${W_{\mathrm {aff}}}$ is a normal subgroup in ${W_{\mathrm {ext}}}$.

We will consider the balanced ‘realization’ of ${W_{\mathrm {aff}}}$ over $\Bbbk$ (in the sense of [Reference Elias and WilliamsonEW16]) defined as follows.

• – The underlying $\Bbbk$-vector space is $\mathbf {t}^{*}$.

• – If $\alpha \in \Phi ^{\mathrm {s}}$ and $s=s_\alpha$, then the ‘root’ $\alpha _s \in \mathbf {t}$ (respectively, ‘coroot’ $\alpha _s^{\vee } \in \mathbf {t}^{*}$) associated with $s$ is the differential of $\alpha ^{\vee }$ (respectively, of $\alpha$).

• – If $\beta \in \Phi ^{+}$ is such that $\beta ^{\vee }$ is a maximal coroot and $s=t_\beta s_\beta$ then the ‘root’ $\alpha _s$ (respectively, ‘coroot’ $\alpha _s^{\vee }$) associated with $s$ is the differential of $-\beta ^{\vee }$ (respectively, of $-\beta$).

This realization is an example of a Cartan realization in the sense of [Reference Achar, Makisumi, Riche and WilliamsonAMRW17, § 10.1]. Our assumption (ii) implies that this realization satisfies the ‘Demazure surjectivity’ condition of [Reference Elias and WilliamsonEW16]. There is an associated action of ${W_{\mathrm {aff}}}$ on $\mathbf {t}^{*}$, which is simply the natural action of $\mathbf {W}$, seen as an action of ${W_{\mathrm {aff}}}$ via the projection ${W_{\mathrm {aff}}} \to \mathbf {W}$.

We will denote by $\mathsf {D}_{\mathrm {BS}}$ the diagrammatic Hecke category defined by Elias–Williamson [Reference Elias and WilliamsonEW16] for the Coxeter system $({W_{\mathrm {aff}}},{S_{\mathrm {aff}}})$ and this choice of realization. (For a discussion of this definition, see also [Reference Achar, Makisumi, Riche and WilliamsonAMRW17, Chap. 2].) The technical conditions necessary for this category to be defined (and well behaved) are somewhat subtle, and not all of them are made explicit in [Reference Elias and WilliamsonEW16]; see [Reference Elias and WilliamsonEW20, § 5] for a detailed discussion of this question. As explained in [Reference Elias and WilliamsonEW20, § 5.1], a sufficient condition (in addition to the fact that the data define a realization satisfying Demazure surjectivity) that ensures that all the results of [Reference Elias and WilliamsonEW16] are applicable is that for any pair of distinct simple reflections $s,t$ such that $st$ has finite order, the restriction of the action to the subgroup generated by $s$ and $t$ is faithful. It follows from [Reference Achar, Makisumi, Riche and WilliamsonAMRW17, Lemma 8.1.1] that this condition is satisfied in our context, except possibly if $p=2$ and $st=ts$. (The assumptions of [Reference Achar, Makisumi, Riche and WilliamsonAMRW17, Lemma 8.1.1] are easily checked by hand for Cartan realizations in good characteristic.) However, in this case $s$ and $t$ act non-trivially by Demazure surjectivity, and $st$ acts non-trivially since $\alpha _s^{\vee }$ and $\alpha _t^{\vee }$ are not collinear thanks to assumption (ii).

The category $\mathsf {D}_{\mathrm {BS}}$ is a $\Bbbk$-linear (non-additive) monoidal category. By definition its objects are pairs $(\underline {w},n)$ where $\underline {w}$ is a word in ${S_{\mathrm {aff}}}$ and $n \in \mathbb {Z}$; the product is given by concatenation of words and addition of integers, and for any words $\underline {w},\underline {w}'$ the direct sum of morphism spaces

\[ \bigoplus_{n \in \mathbb{Z}} {{\rm Hom}}_{\mathsf{D}_{\mathrm{BS}}}((\underline{w},0),(\underline{w}',n)) \]

is a graded bimodule over $R:=\mathscr {O}(\mathbf {t}^{*}) = \mathrm {Sym}(\mathbf {t})$ (where the grading is such that elements in $\mathbf {t}$ have degree $2$). Following usual conventions, the object $(\underline {w},n)$ will rather be denoted $B_{\underline {w}}(n)$. Then there exists a natural ‘grading shift’ autoequivalence of $\mathsf {D}_{\mathrm {BS}}$ such that $(B_{\underline {w}}(n))(1)=B_{\underline {w}}(n+1)$ for any $\underline {w}$ and any $n \in \mathbb {Z}$.

2.2 Abe's incarnation of the Hecke category

For our present purposes we will need a description of $\mathsf {D}_{\mathrm {BS}}$ in terms of $R$-bimodules due to Abe [Reference AbeAbe21], which is close to the definition of Soergel bimodules [Reference SoergelSoe07], and which we now recall. Once again, in order to apply these results one needs some technical assumptions. A sufficient condition (in terms of vanishing of quantum binomial coefficients) for the results of [Reference AbeAbe21] to be applicable is given in [Reference AbeAbe20]. One can check by explicit computation that this condition is automatically satisfied for Cartan realizations.

We will denote by $Q$ the fraction field of $R$. Following [Reference AbeAbe21], we denote by $\mathsf {C}'$ the category defined as follows. Objects are pairs consisting of a graded $R$-bimodule $M$ together with a decomposition

(2.1)\begin{equation} M \otimes_R Q = \bigoplus_{w \in {W_{\mathrm{aff}}}} M_Q^{w} \end{equation}

as $(R,Q)$-bimodules such that:

• – there exist only finitely many elements $w$ such that $M_Q^{w} \neq 0$;

• – for any $w \in {W_{\mathrm {aff}}}$, $r \in R$ and $m \in M_Q^{w}$ we have

  (2.2)\begin{equation} m \cdot r=w(r) \cdot m. \end{equation}

Morphisms in $\mathsf {C}'$ are defined in the obvious way, as morphisms of graded bimodules compatible with the decompositions (2.1). We also denote by $\mathsf {C}$ the full subcategory of $\mathsf {C}'$ whose objects are those whose underlying graded $R$-bimodule $M$ is finitely generated as an $R$-bimodule and flat as a right $R$-module. As explained in [Reference AbeAbe21, Lemma 2.6], the underlying $R$-bimodule of any object in $\mathsf {C}$ is in fact finitely generated as a left and as a right $R$-module; this property shows that the tensor product over $R$ induces in a natural way a monoidal product on $\mathsf {C}$. We also have a ‘grading shift’ autoequivalence of $\mathsf {C}$, which only changes the grading of the underlying graded $R$-bimodule in such a way that $M(1)^{i}=M^{i+1}$.

For $s \in {S_{\mathrm {aff}}}$, we consider the $s$-invariants $R^{s} \subset R$, and the graded $R$-bimodule $B_s^{\mathrm {Bim}} := R \otimes _{R^{s}} R (1)$. This object has a natural ‘lift’ as an object in $\mathsf {C}$, which will also be denoted by $B_s^{\mathrm {Bim}}$ (see [Reference AbeAbe21, § 2.4]).

The following result is a special case of [Reference AbeAbe21, Theorem 5.9]; see also [Reference AbeAbe20, Theorem 1.3].

Theorem 2.2 There exists a canonical fully faithful monoidal functor

\[ \mathsf{D}_{\mathrm{BS}} \to \mathsf{C} \]

sending $B_s$ to $B_s^{\mathrm {Bim}}$ for any $s \in {S_{\mathrm {aff}}}$ and intertwining the grading shifts $(1)$.

It will be convenient to consider also a slight extension of the category $\mathsf {C}$, adapted to the group ${W_{\mathrm {ext}}}$. Namely, the action of ${W_{\mathrm {aff}}}$ on $\mathbf {t}^{*}$ extends in a natural way to ${W_{\mathrm {ext}}}$ (using now the projection ${W_{\mathrm {ext}}} \to \mathbf {W}$). We will denote by $\mathsf {C}'_{\mathrm {ext}}$ the category whose objects are pairs consisting a graded $R$-bimodule $M$ together with a decomposition

\[ M \otimes_R Q = \bigoplus_{w \in {W_{\mathrm{ext}}}} M_Q^{w} \]

as $(R,Q)$-bimodules such that:

• – there exist only finitely many elements $w$ such that $M_Q^{w} \neq 0$;

• – for any $w \in {W_{\mathrm {ext}}}$, $r \in R$ and $m \in M_Q^{w}$ we have $m \cdot r=w(r) \cdot m$,

and where morphisms are defined in the obvious way. We will also denote by $\mathsf {C}_\mathrm {ext}$ the full subcategory of $\mathsf {C}'_\mathrm {ext}$ whose objects are those whose underlying graded $R$-bimodule $M$ is finitely generated as an $R$-bimodule and flat as a right $R$-module. It is clear that $\mathsf {C}'$ is a full subcategory in $\mathsf {C}'_\mathrm {ext}$, that $\mathsf {C}$ is a full subcategory in $\mathsf {C}_\mathrm {ext}$, and that the tensor product $\otimes _R$ defines a monoidal structure on $\mathsf {C}_\mathrm {ext}$.

In addition to the objects $B_s^{\mathrm {Bim}}$ considered above, the category $\mathsf {C}_\mathrm {ext}$ possesses ‘standard’ objects $(\Delta _x : x \in {W_{\mathrm {ext}}})$ defined as follows. For any $x \in {W_{\mathrm {ext}}}$, $\Delta _x$ is isomorphic to $R$ as a graded vector space, and the $R$-bimodule structure is given by

\[ r \cdot m \cdot r' = rmx(r') \]

for $r,r' \in R$ and $m \in \Delta _x$. The decomposition of $\Delta _x \otimes _R Q$ is defined so that this object is concentrated in degree $x$. For any $x,y \in {W_{\mathrm {ext}}}$ we have a canonical isomorphism

(2.3)\begin{equation} \Delta_x \otimes_R \Delta_y \xrightarrow{\sim} \Delta_{xy} \end{equation}

in $\mathsf {C}_\mathrm {ext}$, defined by $m \otimes m' \mapsto mx(m')$.

Lemma 2.4 Let $s,t \in {S_{\mathrm {aff}}}$ and $x \in {W_{\mathrm {ext}}}$ be such that $s=xtx^{-1}$. Then there exists a canonical isomorphism

\[ B_s^{\mathrm{Bim}} \cong \Delta_x \otimes_R B_t^{\mathrm{Bim}} \otimes_R \Delta_{x^{-1}}. \]

Proof. The isomorphism of $R$-bimodules

\[ \Delta_x \otimes_R B_t^{\mathrm{Bim}} \otimes_R \Delta_{x^{-1}} \xrightarrow{\sim} B_s^{\mathrm{Bim}} \]

is defined by

\[ r_1 \otimes (r_2 \otimes r_3) \otimes r_4 \mapsto (r_1x(r_2)) \otimes (x(r_3)x(r_4)). \]

We leave it to the reader to check that this morphism is well defined, and indeed defines an isomorphism in $\mathsf {C}_\mathrm {ext}$.

In § 6 we will also need the following standard claim, for which we refer to [Reference Elias and WilliamsonEW16, § 3.4].

Lemma 2.5 For any $s \in {S_{\mathrm {aff}}}$, there exist exact sequences of $R$-bimodules

\[ \Delta_s \hookrightarrow R \otimes_{R^{s}} R \twoheadrightarrow \Delta_e, \qquad \Delta_e \hookrightarrow R \otimes_{R^{s}} R \twoheadrightarrow \Delta_s. \]

2.3 Universal centralizer and Kostant section

We will denote by $\mathbf {g}_{\mathrm {reg}} \subset \mathbf {g}$ the open subset consisting of regular elements, that is, elements whose centralizer has dimension $\dim (T)$. The ‘regular universal centralizer’ is the affine group scheme

\[ \mathbb{J}_\mathrm{reg} := \mathbf{g}_{\mathrm{reg}} \times_{\mathbf{g}_{\mathrm{reg}} \times \mathbf{g}_{\mathrm{reg}}} (\mathbf{G} \times \mathbf{g}_{\mathrm{reg}}) \]

over $\mathbf {g}_{\mathrm {reg}}$, where the morphism $\mathbf {g}_{\mathrm {reg}} \to \mathbf {g}_{\mathrm {reg}} \times \mathbf {g}_{\mathrm {reg}}$ is the diagonal embedding, and the map $\mathbf {G} \times \mathbf {g}_{\mathrm {reg}} \to \mathbf {g}_{\mathrm {reg}}$ sends $(g,x)$ to $(g \cdot x, x)$. For any $x \in \mathbf {g}_{\mathrm {reg}}$, the fiber of $\mathbb {J}_\mathrm {reg}$ over $x$ is the scheme-theoretic centralizer of $x$ for the adjoint $\mathbf {G}$-action. By construction $\mathbb {J}_\mathrm {reg}$ is a closed subgroup scheme in $\mathbf {G} \times \mathbf {g}_{\mathrm {reg}}$, and as explained in [Reference RicheRic17, Corollary 3.3.6] it is smooth over $\mathbf {g}_{\mathrm {reg}}$. We will also denote by $\mathbf {g}_{\mathrm {reg}}^{*}$ the image of $\mathbf {g}_{\mathrm {reg}}$ under $\kappa$, and by $\mathbb {J}_\mathrm {reg}^{*}$ the smooth affine group scheme over $\mathbf {g}_{\mathrm {reg}}^{*}$ obtained by pushforward from $\mathbb {J}_\mathrm {reg}$. (It is easily seen that these objects do not depend on the choice of $\kappa$.)

There exists a canonical morphism

(2.4)\begin{equation} \mathbf{t}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbb{J}^{*}_\mathrm{reg} \to (\mathbf{t}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{g}_{\mathrm{reg}}^{*}) \times \mathbf{T} \end{equation}

of group schemes over $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {g}_{\mathrm {reg}}^{*}$, whose construction we now explain. Let $\mathbf {n}$ be the Lie algebra of the unipotent radical $\mathbf {U}$ of $\mathbf {B}$. Recall that the Grothendieck resolution is the $\mathbf {G}$-equivariant vector bundle over $\mathbf {G}/\mathbf {B}$ given by

\[ \tilde{\mathbf{g}} := \{(\xi, g\mathbf{B}) \in \mathbf{g}^{*} \times \mathbf{G}/\mathbf{B} \mid \xi_{|g \cdot \mathbf{n}}=0\}. \]

We have natural maps

\[ \pi : \tilde{\mathbf{g}} \to \mathbf{g}^{*}, \quad \vartheta : \tilde{\mathbf{g}} \to \mathbf{t}^{*}. \]

(The morphism $\pi$ is induced by the first projection. The morphism $\vartheta$ sends a pair $(\xi,g\mathbf {B})$ to $\xi _{|g \cdot \mathbf {b}}$, seen as an element in $(g \cdot \mathbf {b} / g \cdot \mathbf {n})^{*} \cong (\mathbf {b}/\mathbf {n})^{*} \cong \mathbf {t}^{*}$, where the first isomorphism is induced by conjugation by the inverse of any representative for the coset $g\mathbf {B}$.) If we denote by $\tilde {\mathbf {g}}_{\mathrm {reg}}$ the preimage of $\mathbf {g}_{\mathrm {reg}}^{*}$ in $\tilde {\mathbf {g}}$, then these maps induce an isomorphism of schemes

(2.5)\begin{equation} \tilde{\mathbf{g}}_{\mathrm{reg}} \xrightarrow{\sim} \mathbf{g}_{\mathrm{reg}}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*}; \end{equation}

see [Reference RicheRic17, Lemma 3.5.3]. Moreover, under this identification, by [Reference RicheRic17, Proposition 3.5.6] the group scheme $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_\mathrm {reg}$ identifies with the universal stabilizer associated with the action of $\mathbf {G}$ on $\tilde {\mathbf {g}}_{\mathrm {reg}}$ (defined by the same procedure as for $\mathbb {J}_\mathrm {reg}$ above), which is such that the fiber over $(\xi,g\mathbf {B})$ is the scheme-theoretic stabilizer of $\xi$ for the action of $g\mathbf {B} g^{-1}$. Now as above in the definition of $\vartheta$, there exists for any $g \in \mathbf {G}$ a canonical isomorphism $g\mathbf {B} g^{-1} / g\mathbf {U} g^{-1} \cong \mathbf {T}$, which allows us to define the wished-for morphism (2.4).

Let $\mathbf {g}_{\mathrm {rs}} \subset \mathbf {g}$ denote the open subset of semisimple regular elements, and set $\mathbf {g}_{\mathrm {rs}}^{*}:=\kappa (\mathbf {g}_{\mathrm {rs}})$. We will denote by $\mathbb {J}_\mathrm {rs}$ (respectively, $\mathbb {J}^{*}_\mathrm {rs}$) the restriction of $\mathbb {J}_\mathrm {reg}$ (respectively, $\mathbb {J}^{*}_\mathrm {reg}$) to $\mathbf {g}_{\mathrm {rs}}$ (respectively, $\mathbf {g}_{\mathrm {rs}}^{*}$). Recall that the adjoint quotient $\mathbf {g}/\mathbf {G}$ identifies canonically with $\mathbf {t}/\mathbf {W}$ (see [Reference Bouthier and CesnaviciusBC19, § 4.1]); as a consequence, under our assumptions the coadjoint quotient $\mathbf {g}^{*}/\mathbf {G}$ identifies canonically with $\mathbf {t}^{*}/\mathbf {W}$.

Lemma 2.6 The morphism (2.4) restricts to an isomorphism

\[ \mathbb{J}^{*}_\mathrm{rs} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*} \xrightarrow{\sim} \mathbf{T} \times (\mathbf{g}_{\mathrm{rs}}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*}) \]

of group schemes over $\mathbf {g}_{\mathrm {rs}}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}$.

Proof. It is sufficient to prove the analogous statement for $\mathbf {g}$ in place of $\mathbf {g}^{*}$. If we denote by $\tilde {\mathbf {g}}_\mathrm {rs}$ the inverse image of $\mathbf {g}_{\mathrm {rs}}$ under $\pi$, then by [Reference JantzenJan04, Lemma 13.4] there exists a canonical isomorphism

\[ \mathbf{G} \times^{\mathbf{T}} \mathbf{t}_\mathrm{rs} \xrightarrow{\sim} \tilde{\mathbf{g}}_\mathrm{rs}, \]

where $\mathbf {t}_\mathrm {rs} := \mathbf {t} \cap \mathbf {g}_\mathrm {rs}$. From the comments above and the compatibility of universal stabilizers with open embeddings, $\mathbf {t} \times _{\mathbf {t}/\mathbf {W}} \mathbb {J}_\mathrm {rs}$ identifies with the universal stabilizer associated with the action of $\mathbf {G}$ on $\tilde {\mathbf {g}}_\mathrm {rs}$. Since the latter scheme identifies with $\mathbf {G} \times ^{\mathbf{T}} \mathbf {t}_\mathrm {rs} = \mathbf {G}/\mathbf {T} \times \mathbf {t}_\mathrm {rs}$, the universal stabilizer identifies with $\mathbf {T} \times \tilde {\mathbf {g}}_\mathrm {rs}$, as desired.

Let us choose a Kostant section to the adjoint quotient, that is, a closed subscheme $\mathbf {S} \subset \mathbf {g}$ contained in $\mathbf {g}_{\mathrm {reg}}$ and such that the composition $\mathbf {S} \hookrightarrow \mathbf {g} \to \mathbf {g}/\mathbf {G}$ (where the second map is the adjoint quotient morphism) is an isomorphism. (For a construction of such a section in the present generality, see [Reference RicheRic17, § 3].) We will denote by $\mathbf {S}^{*}$ the image of $\mathbf {S}$ under $\kappa$, so that the composition $\mathbf {S}^{*} \hookrightarrow \mathbf {g}^{*} \to \mathbf {g}^{*}/\mathbf {G}$ is an isomorphism, and by $\mathbb {J}^{*}_\mathbf {S}$ the restriction of $\mathbb {J}_\mathrm {reg}^{*}$ to $\mathbf {S}^{*}$ (a closed subgroup scheme of $\mathbf {G} \times \mathbf {S}^{*}$, smooth over $\mathbf {S}^{*}$).

As explained in [Reference Mautner and RicheMR18, § 4.4], for example, there exists a natural action of the multiplicative group $\mathbb {G}_{\mathrm {m}}$ on $\mathbf {S}^{*}$ such that the isomorphism $\mathbf {S}^{*} \xrightarrow {\sim } \mathbf {g}^{*}/\mathbf {G}$ is $\mathbb {G}_{\mathrm {m}}$-equivariant, where $t \in \Bbbk ^{\times }$ acts on $\mathbf {g}^{*}$ by multiplication by $t^{-2}$, and on $\mathbf {g}^{*}/\mathbf {G}$ by the induced action. The isomorphism $\mathbf {t}^{*}/\mathbf {W} \xrightarrow {\sim } \mathbf {g}^{*}/\mathbf {G}$ is also $\mathbb {G}_{\mathrm {m}}$-equivariant, where the action on $\mathbf {t}^{*}/\mathbf {W}$ is induced by the action on $\mathbf {t}^{*}$ where $t \in \Bbbk ^{\times }$ acts by multiplication by $t^{-2}$. As explained in [Reference Mautner and RicheMR18, § 4.5, p. 2302] there exists a natural $\mathbb {G}_{\mathrm {m}}$-action on $\mathbb {J}^{*}_{\mathbf {S}}$ such that the structure morphism $\mathbb {J}^{*}_{\mathbf {S}} \to \mathbf {S}^{*}$, the multiplication map $\mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {S}^{*}} \mathbb {J}^{*}_{\mathbf {S}} \to \mathbb {J}^{*}_{\mathbf {S}}$ and the inversion morphism $\mathbb {J}^{*}_{\mathbf {S}} \to \mathbb {J}^{*}_{\mathbf {S}}$ are $\mathbb {G}_{\mathrm {m}}$-equivariant.

2.4 Representations of the universal centralizer and Abe's category

The actions of $\mathbb {G}_{\mathrm {m}}$ on $\mathbf {t}^{*}$ and $\mathbf {t}^{*}/\mathbf {W}$ considered in § 2.3 provide an action on the fiber product $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}$. Let us now consider the category

\[ \mathsf{Rep}^{\mathbb{G}_{\mathrm{m}}}(\mathbf{t}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbb{J}^{*}_{\mathbf{S}} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*}) \]

of $\mathbb {G}_{\mathrm {m}}$-equivariant coherent representations of the smooth affine group scheme

\[ \mathbf{t}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbb{J}^{*}_{\mathbf{S}} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*} \]

over $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}$ (where the morphism $\mathbb {J}^{*}_\mathbf {S} \to \mathbf {t}^{*}/\mathbf {W}$ is obtained via the identification $\mathbf {S}^{*} \xrightarrow {\sim } \mathbf {t}^{*}/\mathbf {W}$), that is, $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}$-modules equipped with a structure of $\mathbb {G}_{\mathrm {m}}$-equivariant coherent sheaf on $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}$, such that the action map is $\mathbb {G}_{\mathrm {m}}$-equivariant. Since $R$ is finite over $R^{\mathbf{W}}$, this category admits a natural convolution product $\star$, such that the $\mathscr {O}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$-module underlying the product $M \star N$ is the tensor product $M \otimes _{R} N$ (where $R=\mathscr {O}(\mathbf {t}^{*})$ acts on $M$ via the second projection $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*} \to \mathbf {t}^{*}$ and on $N$ via the first projection $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*} \to \mathbf {t}^{*}$). In this way, $(\mathsf {Rep}^{\mathbb {G}_{\mathrm {m}}}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}), \star )$ is a monoidal category. We will denote by

\[ \mathsf{Rep}_{\mathrm{fl}}^{\mathbb{G}_{\mathrm{m}}}(\mathbf{t}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbb{J}^{*}_{\mathbf{S}} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*}) \]

the full subcategory of $\mathsf {Rep}^{\mathbb {G}_{\mathrm {m}}}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$ whose objects are the representations whose underlying coherent sheaves are flat with respect to the second projection $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*} \to \mathbf {t}^{*}$. It is not difficult to check that this subcategory is stable under $\star$, hence also admits a canonical monoidal category structure.

Proposition 2.7 There exists a canonical fully faithful monoidal functor

\[ \big( \mathsf{Rep}_{\mathrm{fl}}^{\mathbb{G}_{\mathrm{m}}}(\mathbf{t}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbb{J}^{*}_{\mathbf{S}} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*}),\star \big) \to (\mathsf{C}_\mathrm{ext}, \otimes_R). \]

Proof. We start by constructing a functor

(2.6)\begin{equation} \mathsf{Rep}^{\mathbb{G}_{\mathrm{m}}}(\mathbf{t}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbb{J}^{*}_{\mathbf{S}} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*}) \to \mathsf{C}'_{\mathrm{ext}}. \end{equation}

By definition, any object in $\mathsf {Rep}^{\mathbb {G}_{\mathrm {m}}}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$ is in particular a $\mathbb {G}_{\mathrm {m}}$-equivariant coherent sheaf on $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}$, hence can be seen as a graded $R$-bimodule. To equip this graded bimodule with the structure of an object in $\mathsf {C}'_\mathrm {ext}$, we must provide a decomposition of its tensor product with $Q$ parametrized by ${W_{\mathrm {ext}}}$. In fact, we will provide such a decomposition for its tensor product with $\mathscr {O}(\mathbf {t}^{*}_\mathrm {rs})$, where $\mathbf {t}^{*}_\mathrm {rs} := \mathbf {t}^{*} \cap \mathbf {g}^{*}_\mathrm {rs}$ (which is sufficient since $Q$ is a further localization of $\mathscr {O}(\mathbf {t}^{*}_\mathrm {rs})$).

First, the open subset $\mathbf {t}^{*}_{\mathrm {rs}} \subset \mathbf {t}^{*}$ is the complement of the kernels of the differentials of the coroots. This open subset is stable under the action of $\mathbf {W}$, and the restriction of this action is free; see [Reference RicheRic17, Lemma 2.3.3]. In particular, we have an open subset $\mathbf {t}^{*}_{\mathrm {rs}} / \mathbf {W} \subset \mathbf {t}^{*}/\mathbf {W}$, the morphism $\mathbf {t}^{*}_\mathrm {rs} \to \mathbf {t}^{*}_\mathrm {rs}/\mathbf {W}$ is étale, and the map $(w,x) \mapsto (x,w(x))$ induces an isomorphism of schemes

\[ \mathbf{W} \times \mathbf{t}^{*}_\mathrm{rs} \xrightarrow{\sim} \mathbf{t}^{*}_\mathrm{rs} \times_{\mathbf{t}^{*}_\mathrm{rs}/\mathbf{W}} \mathbf{t}^{*}_\mathrm{rs}; \]

see [SGA1, Exp. V, § 2]. As a consequence, for any coherent sheaf $\mathscr {F}$ on $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}$, the tensor product

\[ \Gamma(\mathbf{t}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*}, \mathscr{F}) \otimes_R \mathscr{O}(\mathbf{t}^{*}_\mathrm{rs}) \]

admits a canonical decomposition (as an $\mathscr {O}(\mathbf {t}^{*}_\mathrm {rs})$-bimodule) parametrized by $\mathbf {W}$, such that the action on the factor corresponding to $w \in \mathbf {W}$ factors through the quotient

\[ \mathscr{O}(\mathbf{t}_\mathrm{rs}^{*} \times \mathbf{t}^{*}_\mathrm{rs}) \twoheadrightarrow \mathscr{O}(\mathrm{Gr}(w,\mathbf{t}^{*}_\mathrm{rs})) \]

(where on the right-hand side $\mathrm {Gr}(w,\mathbf {t}^{*}_\mathrm {rs})$ denotes the graph of $w$ acting on $\mathbf {t}^{*}_\mathrm {rs}$), that is, satisfies the condition in (2.2).

Next, let us explain how this decomposition can be refined if $\mathscr {F}$ belongs to $\mathsf {Rep}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$. For this, we consider the restriction $\mathbf {M}_w$ of $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}$ to $\mathrm {Gr}(w,\mathbf {t}^{*}_\mathrm {rs})$. Identifying the latter subscheme with $\mathbf {t}^{*}_\mathrm {rs}$ via the first projection and using Lemma 2.6, we obtain a canonical isomorphism of group schemes

\[ \mathbf{M}_w \xrightarrow{\sim} \mathbf{t}^{*}_\mathrm{rs} \times \mathbf{T}. \]

This means that the category of representations of $\mathbf {M}_w$ on coherent sheaves on $\mathrm {Gr}(w,\mathbf {t}^{*}_\mathrm {rs})$ is canonically equivalent to the category of $\mathbf {X}$-graded coherent sheaves on $\mathbf {t}^{*}_\mathrm {rs}$. Starting with an object $\mathscr {F}$ in $\mathsf {Rep}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$, we therefore obtain a decomposition of $\Gamma (\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}, \mathscr {F}) \otimes _R \mathscr {O}(\mathbf {t}^{*}_\mathrm {rs})$ parametrized by ${W_{\mathrm {ext}}}$ by defining, for $\lambda \in \mathbf {X}$ and $w \in \mathbf {W}$, the summand associated with $t_\lambda w$ as the $\lambda$-graded part in the summand associated with $w$ (which is a representation of $\mathbf {M}_w$). This finishes the description of the functor (2.6).

It is clear from construction that this functor sends objects in $\mathsf {Rep}^{\mathbb {G}_{\mathrm {m}}}_{\mathrm {fl}}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$ to objects in $\mathsf {C}_{\mathrm {ext}}$, which therefore provides the functor of the statement. This functor is also easily seen to be monoidal. Let us now explain why it is fully faithful. Consider $\mathscr {F},\mathscr {G}$ in $\mathsf {Rep}^{\mathbb {G}_{\mathrm {m}}}_{\mathrm {fl}}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$, and denote their images by $M,N$—so that the underlying graded bimodule of $M$ (respectively, $N$) is $\Gamma (\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*},\mathscr {F})$ (respectively, $\Gamma (\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*},\mathscr {G})$). By construction, morphisms in $\mathsf {C}_{\mathrm {ext}}$ from $M$ to $N$ are morphisms of graded bimodules from $\Gamma (\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*},\mathscr {F})$ to $\Gamma (\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*},\mathscr {G})$ whose restriction to $\mathbf {t}^{*} \times \{\eta \}$ (where $\eta$ is the generic point of $\mathbf {t}^{*}$) commutes with the action of the restriction of $\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*}$. Now, since by assumption $\Gamma (\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*},\mathscr {G})$ is flat as a right $R$-module and $\mathscr {O}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$ is flat over $\mathscr {O}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$, such a morphism is automatically a morphism of $\mathscr {O}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$-comodules. This proves the desired full faithfulness.

For $w \in {W_{\mathrm {ext}}}$, we will denote by $\Delta _w^{\mathbb {J}}$ the unique object in $\mathsf {Rep}_{\mathrm {fl}}^{\mathbb {G}_{\mathrm {m}}}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$ which is sent to $\Delta _w$.

2.5 Representations of the universal centralizer and the Hecke category

By Proposition 2.7 the category $\mathsf {Rep}_{\mathrm {fl}}^{\mathbb {G}_{\mathrm {m}}}(\mathbf {t}^{*} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbb {J}^{*}_{\mathbf {S}} \times _{\mathbf {t}^{*}/\mathbf {W}} \mathbf {t}^{*})$ can be seen as a full monoidal subcategory in Abe's category $\mathsf {C}_\mathrm {ext}$, and by Theorem 2.2 the same is true for the Hecke category $\mathsf {D}_{\mathrm {BS}}$. We now investigate the relation between these two subcategories.

From this lemma we deduce the following claim, which will be crucial for our constructions in § 6.

Theorem 2.10 There exists a canonical fully faithful monoidal functor

\[ \mathsf{D}_{\mathrm{BS}} \to \mathsf{Rep}_{\mathrm{fl}}^{\mathbb{G}_{\mathrm{m}}}(\mathbf{t}^{*} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbb{J}^{*}_{\mathbf{S}} \times_{\mathbf{t}^{*}/\mathbf{W}} \mathbf{t}^{*}). \]

3.1 Weights

From now on we assume that $p>0$. We consider a simply connected semisimple algebraic group $G$ over $\Bbbk$, and its category

\[ \mathsf{Rep}(G) \]

of finite-dimensional algebraic representations.

For a $\Bbbk$-scheme $X$ we will denote by $X^{(1)}$ the associated Frobenius twist, defined as the fiber product $X^{(1)}:=X \times _{\mathrm {Spec}(\Bbbk )} \,\mathrm {Spec}(\Bbbk )$, where the morphism $\mathrm {Spec}(\Bbbk ) \to \mathrm {Spec}(\Bbbk )$ is associated with the map $x \mapsto x^{p}$. (The projection $X^{(1)} \to X$ is an isomorphism of $\mathbb {F}_p$-schemes, but not of $\Bbbk$-schemes.) We will assume that

\[ p \text{ is very good for } G. \]

Then the group $\mathbf {G} := G^{(1)}$ satisfies the assumptions of § 2. We will denote by $\mathrm {Fr} : G \to \mathbf {G}$ the Frobenius morphism of $G$, and will use the same notation for its restriction to the various subgroups considered below.

The subgroups $\mathbf {B}$, $\mathbf {T}$, $\mathbf {U}$ of $\mathbf {G}$, when seen as subschemes in $G$, determine subgroups $B$, $T$, $U$ whose Frobenius twists are $\mathbf {B}$, $\mathbf {T}$, $\mathbf {U}$, respectively. We will denote by $\mathfrak {g}$, $\mathfrak {b}$, $\mathfrak {t}$, $\mathfrak {n}$ the respective Lie algebras of $G$, $B$, $T$, $U$ (so that $\mathbf {g}=\mathfrak {g}^{(1)}$, and similarly for $B$, $U$, $T$), and by $W$ the Weyl group of $(G,T)$. We set $\mathbb {X}:=X^{*}(T)$, and denote by $\mathfrak {R} \subset \mathbb {X}$ the root system of $(G,T)$. The choice of $B$ determines a system of positive roots $\mathfrak {R}^{+} \subset \mathfrak {R}$, chosen as the $T$-weights in $\mathfrak {g}/\mathfrak {b}$. We will denote by $\mathfrak {R}^{\mathrm {s}} \subset \mathfrak {R}$ the corresponding subset of simple roots, and by $\rho \in \mathbb {X}$ the half-sum of the positive roots. We also set $\mathbb {X}^{\vee } := X_*(T)$, and denote by $\mathfrak {R}^{\vee } \subset \mathbb {X}^{\vee }$ the coroot system. The canonical bijection $\mathfrak {R} \xrightarrow {\sim } \mathfrak {R}^{\vee }$ will be denoted as usual $\alpha \mapsto \alpha ^{\vee }$.

The Frobenius morphism $\mathrm {Fr}$ induces an isomorphism

\[ N_G(T)/T \xrightarrow{\sim} N_{\mathbf{G}}(\mathbf{T})/\mathbf{T}, \]

which allows us to identify the Weyl group $\mathbf {W}$ of $\mathbf {G}$ with $W$. It is a standard fact that the morphism from $\mathbf {X}=X^{*}(T^{(1)})$ to $\mathbb {X}$ induced by $\mathrm {Fr} : T \to \mathbf {T}$ is injective, and that its image is $p \cdot \mathbb {X}$, which allows us to identify $\mathbf {X}$ with $p \cdot \mathbb {X}$. The identification $\mathbf {X} = p \cdot \mathbb {X}$ is $W$-equivariant, and the root system $\Phi$ of $(\mathbf {G},\mathbf {T})$ is $\Phi = \{p \cdot \alpha : \alpha \in \mathfrak {R}\}$; similarly, we have $\Phi ^{+}=p\mathfrak {R}^{+}$ and $\Phi ^{\mathrm {s}}=p\mathfrak {R}^{\mathrm {s}}$. In particular, the affine Weyl group ${W_{\mathrm {aff}}}$ of § 2.1 identifies with $W \ltimes (p \mathbb {Z}\mathfrak {R})$, and the extended affine Weyl group ${W_{\mathrm {ext}}}$ identifies with $W \ltimes (p \cdot \mathbb {X})$. Recall also our subset of Coxeter generators ${S_{\mathrm {aff}}} \subset {W_{\mathrm {aff}}}$. The subgroup $W \subset {W_{\mathrm {aff}}}$ is a parabolic subgroup; its longest element will be denoted (as usual) by $w_0$. We will consider the ‘dot’ action of ${W_{\mathrm {ext}}}$ (or its subgroup ${W_{\mathrm {aff}}}$) on $\mathbb {X}$ defined by

\[ (t_\mu w) \bullet \lambda = w(\lambda+\rho)-\rho+\mu \]

for $\mu \in p\mathbb {X}$, $w \in W$ and $\lambda \in \mathbb {X}$.

Given a character $\lambda \in \mathbb {X}$, we will denote by ${{\bar {\lambda }}} \in \mathfrak {t}^{*}$ the differential of $\lambda$. We set

\[ \mathfrak{t}^{*}_{\mathbb{F}_p} := \{{{\bar{\lambda}}} : \lambda \in \mathbb{X}\} \subset \mathfrak{t}^{*}. \]

In this way, the map $\lambda \mapsto {{\bar {\lambda }}}$ induces an isomorphism of abelian groups

\[ \mathbb{X}/p\mathbb{X} \xrightarrow{\sim} \mathfrak{t}^{*}_{\mathbb{F}_p}. \]

(In particular, $\mathfrak {t}^{*}_{\mathbb {F}_p}$ is finite.)

The group $W$ naturally acts on $\mathfrak {t}^{*}$. We also have a ‘dot’ action of $W$ on $\mathfrak {t}^{*}$, defined by

\[ w \bullet \xi := w(\xi+\bar{\rho})-\bar{\rho}. \]

With this definition the map $\mathbb {X} \to \mathfrak {t}^{*}$ sending $\lambda$ to ${{\bar {\lambda }}}$ is ${W_{\mathrm {ext}}}$-equivariant, where ${W_{\mathrm {ext}}}$ acts on $\mathbb {X}$ via the dot action and on $\mathfrak {t}^{*}$ via the projection ${W_{\mathrm {ext}}} \to W$ and the dot action of $W$ on $\mathfrak {t}^{*}$. This observation legitimates the use of the same notation for these actions. It also shows that the subset $\mathfrak {t}^{*}_{\mathbb {F}_p} \subset \mathfrak {t}^{*}$ is stable under the dot action of $W$. Below we will consider the quotient $\mathfrak {t}^{*}/(W,\bullet )$ of the dot action of $W$ on $\mathfrak {t}^{*}$. For $\lambda \in \mathbb {X}$, we will denote by ${{\tilde {\lambda }}}$ the image of ${{\bar {\lambda }}}$ in $\mathfrak {t}^{*}/(W,\bullet )$.

As mentioned above, our assumption that $p$ is very good for $G$ implies in particular that the quotient $\mathbb {X}/\mathbb {Z}\mathfrak {R}$ has no $p$-torsion, or in other words that

(3.1)\begin{equation} \mathbb{Z}\mathfrak{R} \cap p\mathbb{X} = p\mathbb{Z}\mathfrak{R}. \end{equation}

This equality has the following consequences.

Proof. (i) Since $W \bullet \lambda \subset \lambda +\mathbb {Z}\mathfrak {R}$, we have

\[ ({W_{\mathrm{ext}}} \bullet \lambda) \cap (\lambda+\mathbb{Z}\mathfrak{R}) = (W \bullet \lambda + p\mathbb{X}) \cap (\lambda+\mathbb{Z}\mathfrak{R}) = W \bullet \lambda + (\mathbb{Z}\mathfrak{R} \cap p\mathbb{X}). \]

In view of (3.1), the right-hand side equals $W \bullet \lambda + p\mathbb {Z}\mathfrak {R} = {W_{\mathrm {aff}}} \bullet \lambda$, as desired.

(ii) For $w \in W$ we have

\[ w \bullet {{\bar{\lambda}}} = \overline{w \bullet \lambda}, \]

so that $w \bullet {{\bar {\lambda }}}={{\bar {\lambda }}}$ if and only if $w \bullet \lambda \in \lambda +p\mathbb {X}$. Since $w \bullet \lambda \in \lambda +\mathbb {Z}\mathfrak {R}$, as above this condition is equivalent to $w \bullet \lambda \in \lambda +p\mathbb {Z}\mathfrak {R}$, that is, to the existence of $\mu \in p\mathbb {Z}\mathfrak {R}$ such that $t_\mu w \in {W_{\mathrm {aff}}}$ stabilizes $\lambda$.

For any subset $I \subset \mathfrak {R}^{\mathrm {s}}$, we will denote by $W_I \subset W$ the subgroup generated by the reflections $\{s_\alpha : \alpha \in I\}$. Recall that an element of $\mathbb {X}$ is called regular if its stabilizer in ${W_{\mathrm {aff}}}$ (for the dot action) is trivial. As a consequence of Lemma 3.1, we obtain, in particular, the following claim.

Lemma 3.2 Let $\lambda \in \mathbb {X}$, and assume that the stabilizer of $\lambda$ for the dot action of ${W_{\mathrm {aff}}}$ is $W_I$. Then the morphism

\[ \mathfrak{t}^{*} / (W_I,\bullet) \to \mathfrak{t}^{*}/(W,\bullet) \]

induced by the quotient morphism $\mathfrak {t}^{*} \to \mathfrak {t}^{*}/(W,\bullet )$ is étale at the image of ${{\bar {\lambda }}}$. In particular, if $\lambda$ is regular then the quotient morphism $\mathfrak {t}^{*} \to \mathfrak {t}^{*}/(W,\bullet )$ is étale at ${{\bar {\lambda }}}$.

3.2 The center of the enveloping algebra

Consider the universal enveloping algebra $\mathcal {U} \mathfrak {g}$ of $\mathfrak {g}$. Its center $Z(\mathcal {U}\mathfrak {g})$ can be described as follows. We set

\[ {Z_{\mathrm{HC}}}:=(\mathcal{U}\mathfrak{g})^{G}. \]

(Here, the subscript ‘$\mathrm {HC}$’ stands for Harish-Chandra.) Next, as the Lie algebra of an algebraic group over a field of characteristic $p$, $\mathfrak {g}$ admits a ‘restricted $p$th power’ operation $x \mapsto x^{[p]}$, which stabilizes the Lie algebra of any algebraic subgroup of $G$. We will denote by

\[ {Z_{\mathrm{Fr}}} \]

the $\Bbbk$-subalgebra of $\mathcal {U}\mathfrak {g}$ generated by the elements of the form $x^{p}-x^{[p]}$ for $x \in \mathfrak {g}$. Then by [Reference Mirković and RumyninMR99, Theorem 2] multiplication induces an isomorphism

\[ {Z_{\mathrm{Fr}}} \otimes_{{Z_{\mathrm{Fr}}} \cap {Z_{\mathrm{HC}}}} {Z_{\mathrm{HC}}} \xrightarrow{\sim} Z(\mathcal{U}\mathfrak{g}). \]

It is well known that $\mathcal {U} \mathfrak {g}$ is finite as a ${Z_{\mathrm {Fr}}}$-algebra (hence a fortiori as a $Z(\mathcal {U} \mathfrak {g})$-algebra).

These central subalgebras can be described geometrically as follows. It is well known that the map $x \mapsto x^{p}-x^{[p]}$ induces a $\Bbbk$-algebra isomorphism

(3.2)\begin{equation} \mathscr{O}(\mathfrak{g}^{*(1)}) \xrightarrow{\sim} {Z_{\mathrm{Fr}}}. \end{equation}

We also have ${Z_{\mathrm {Fr}}} \cap {Z_{\mathrm {HC}}} = ({Z_{\mathrm {Fr}}})^{G}$, and the $G$-action on $\mathfrak {g}^{*(1)}$ factors through the Frobenius morphism $\mathrm {Fr}$, so that we obtain an isomorphism

\[ \mathscr{O}(\mathfrak{g}^{*}{}^{(1)} / G^{(1)}) \xrightarrow{\sim} {Z_{\mathrm{Fr}}} \cap {Z_{\mathrm{HC}}}. \]

On the other hand, the ‘Harish-Chandra isomorphism’ provides a $\Bbbk$-algebra isomorphism

(3.3)\begin{equation} \mathscr{O}(\mathfrak{t}^{*}/(W,\bullet)) \xrightarrow{\sim} {Z_{\mathrm{HC}}}, \end{equation}

see [Reference Mirković and RumyninMR99, Theorem 1(2)].

The Artin–Schreier morphism

\[ \mathrm{AS} : \mathfrak{t}^{*} \to \mathfrak{t}^{*(1)} \]

is the morphism associated with the algebra map $\mathscr {O}(\mathfrak {t}^{*(1)}) \to \mathscr {O}(\mathfrak {t}^{*})$ defined by $h \mapsto h^{p}-h^{[p]}$ for $h \in \mathfrak {t}$. It is well known that $\mathrm {AS}$ is a Galois covering with Galois group $\mathfrak {t}^{*}_{\mathbb {F}_p}$ (acting on $\mathfrak {t}^{*}$ via addition). The morphism $\mathrm {AS}$ is $W$-equivariant, where $W$ acts on $\mathfrak {t}^{*}$ via the dot action and on $\mathfrak {t}^{*(1)}$ via the natural action. It therefore induces a morphism

(3.4)\begin{equation} \mathfrak{t}^{*}/(W,\bullet) \to \mathfrak{t}^{*}{}^{(1)} / W. \end{equation}

Recall the Chevalley isomorphism

\[ \mathfrak{t}^{*}{}^{(1)} / W \xrightarrow{\sim} \mathfrak{g}^{*}{}^{(1)} / G^{(1)} \]

already encountered in § 2.3. Under this identification, the embedding ${Z_{\mathrm {Fr}}} \cap {Z_{\mathrm {HC}}} \hookrightarrow {Z_{\mathrm {HC}}}$ is induced by (3.4).

Combining all these descriptions, and setting

\[ \mathfrak{C}:= \mathfrak{g}^{*(1)} \times_{\mathfrak{t}^{*(1)} / W} \mathfrak{t}^{*}/(W,\bullet), \]

we therefore obtain a $\Bbbk$-algebra isomorphism

\[ \mathscr{O}(\mathfrak{C}) \xrightarrow{\sim} Z(\mathcal{U} \mathfrak{g}), \]

see [Reference Mirković and RumyninMR99, Corollary 3].

Using this identification one can consider $\mathcal {U} \mathfrak {g}$ as an $\mathscr {O}(\mathfrak {C})$-algebra. The $G$-action on $\mathfrak {C}$ induced by the adjoint $G$-action on $\mathcal {U} \mathfrak {g}$ is the action obtained by pullback via the Frobenius morphism $\mathrm {Fr}$ of the $G^{(1)}$-action on $\mathfrak {C}$ induced by the coadjoint $G^{(1)}$-action on $\mathfrak {g}^{*(1)}$. Using this action, one can therefore see $\mathcal {U} \mathfrak {g}$ as a $G$-equivariant $\mathscr {O}(\mathfrak {C})$-algebra.

3.3 Central reductions

In view of (3.2), the maximal ideals in ${Z_{\mathrm {Fr}}}$ are in a canonical bijection with elements in $\mathfrak {g}^{*(1)}$. Given $\eta \in \mathfrak {g}^{*(1)}$, we will denote by $\mathfrak {m}_\eta \subset {Z_{\mathrm {Fr}}}$ the corresponding maximal ideal, and set

\[ \mathcal{U}_{\eta}\mathfrak{g} := \mathcal{U} \mathfrak{g} / \mathfrak{m}_\eta \cdot \mathcal{U} \mathfrak{g}. \]

Similarly, in view of (3.3) the maximal ideals in ${Z_{\mathrm {HC}}}$ are in a canonical bijection with closed points in $\mathfrak {t}^{*}/(W,\bullet )$, that is, with $(W,\bullet )$-orbits in $\mathfrak {t}^{*}$. Given a closed point $\xi \in \mathfrak {t}^{*}/(W,\bullet )$, we will denote by $\mathfrak {m}^{\xi } \subset {Z_{\mathrm {HC}}}$ the corresponding maximal ideal, and set

\[ \mathcal{U}^{\xi} \mathfrak{g} := \mathcal{U} \mathfrak{g} / \mathfrak{m}^{\xi} \cdot \mathcal{U} \mathfrak{g}. \]

If $\eta$ and $\xi$ have the same image in $\mathfrak {t}^{*(1)}/W$, then $\mathfrak {m}_\eta \cdot Z(\mathcal {U} \mathfrak {g}) + \mathfrak {m}^{\xi } \cdot Z(\mathcal {U} \mathfrak {g})$ is a maximal ideal in $Z(\mathcal {U} \mathfrak {g})$, and we can also set

\[ \mathcal{U}_{\eta}^{\xi}\mathfrak{g} := \mathcal{U} \mathfrak{g} / (\mathfrak{m}_\eta \cdot \mathcal{U} \mathfrak{g} + \mathfrak{m}^{\xi} \cdot \mathcal{U} \mathfrak{g}). \]

In the cases we will encounter more specifically below, the point $\xi$ will often be the image $\tilde {\lambda }$ of the differential of a character $\lambda \in \mathbb {X}$. In this setting we will write $\mathfrak {m}^{\lambda }$, $\mathcal {U}^{\lambda } \mathfrak {g}$ and $\mathcal {U}_{\eta }^{\lambda }\mathfrak {g}$ instead of $\mathfrak {m}^{\tilde {\lambda }}$, $\mathcal {U}^{\tilde {\lambda }} \mathfrak {g}$ and $\mathcal {U}_{\eta }^{\tilde {\lambda }}\mathfrak {g}$. The image of any element of $\mathfrak {t}^{*}_{\mathbb {F}_p}$ under the Artin–Schreier map is $0$; therefore, if we denote by

\[ \mathcal{N}^{*} \subset \mathfrak{g}^{*} \]

the preimage under the coadjoint morphism $\mathfrak {g}^{*} \to \mathfrak {t}^{*}/W$ of the image of $0$, then, given any $\lambda \in \mathbb {X}$, the elements $\eta \in \mathfrak {g}^{*(1)}$ whose image in $\mathfrak {t}^{*(1)}/W$ coincides with that of $\tilde {\lambda }$ are exactly those in $\mathcal {N}^{*(1)}$.

3.4 Harish-Chandra bimodules

We will denote by $\mathsf {HC}$ the category whose objects are the $\mathcal {U} \mathfrak {g}$-bimodules $V$ endowed with an (algebraic) action of $G$ which satisfy the following conditions.

1. (i) The action morphisms $\mathcal {U} \mathfrak {g} \otimes V \to V$ and $V \otimes \mathcal {U} \mathfrak {g} \to V$ are $G$-equivariant (for the diagonal actions on $\mathcal {U} \mathfrak {g} \otimes V$ and $V \otimes \mathcal {U} \mathfrak {g}$).

2. (ii) The $\mathfrak {g}$-action on $V$ obtained by differentiating the $G$-action is given by $(x,v) \mapsto x \cdot v - v \cdot x$.

3. (iii) $V$ is finitely generated both as a left and as a right $\mathcal {U} \mathfrak {g}$-module.

Morphisms in the category $\mathsf {HC}$ are morphisms of bimodules which also commute with the $G$-actions. Objects in this category are called Harish-Chandra bimodules. It is easily seen that the tensor product $\otimes _{\mathcal {U} \mathfrak {g}}$ of bimodules endows $\mathsf {HC}$ with the structure of a monoidal category, where the $G$-action on the tensor product is the diagonal action.

If $M$ belongs to $\mathsf {HC}$, the $\mathcal {U} \mathfrak {g}$-action obtained by differentiating the $G$-action must vanish on ${Z_{\mathrm {Fr}}} \cap (\mathfrak {g} \cdot \mathcal {U} \mathfrak {g})$. In view of condition (ii) above, this implies that the two actions of ${Z_{\mathrm {Fr}}}$ on $M$ obtained by restriction of the left and right $\mathcal {U} \mathfrak {g}$-actions coincide; in other words, the action of $\mathcal {U} \mathfrak {g} \otimes _\Bbbk \mathcal {U} \mathfrak {g}^{\mathrm {op}}$ on $M$ must factor through an action of $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$. However, the two actions of ${Z_{\mathrm {HC}}}$ on a Harish-Chandra bimodule might differ. Note that $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$ is in a natural way a finite algebra over the commutative ring

\[ \mathcal{Z} := Z(\mathcal{U} \mathfrak{g}) \otimes_{{Z_{\mathrm{Fr}}}} Z(\mathcal{U} \mathfrak{g}) = {Z_{\mathrm{HC}}} \otimes_{{Z_{\mathrm{Fr}}} \cap {Z_{\mathrm{HC}}}} {Z_{\mathrm{Fr}}} \otimes_{{Z_{\mathrm{Fr}}} \cap {Z_{\mathrm{HC}}}} {Z_{\mathrm{HC}}} \cong \mathscr{O}(\mathfrak{C} \times_{\mathfrak{g}^{*(1)}} \mathfrak{C}). \]

Note also that since $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$ is finitely generated both as a left and as a right $\mathcal {U} \mathfrak {g}$-module, condition (iii) above can be equivalently replaced by the condition that the object is finitely generated as a $\mathcal {U} \mathfrak {g}$-bimodule (or as a left $\mathcal {U} \mathfrak {g}$-module, or as a right $\mathcal {U} \mathfrak {g}$-module). It is easily seen that forgetting the right (respectively, left) action of $\mathcal {U} \mathfrak {g}$ defines an equivalence of categories

(3.5)\begin{equation} \mathsf{HC} \xrightarrow{\sim} \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g}) \quad (\text{respectively, } \mathsf{HC} \xrightarrow{\sim} \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g}^{\mathrm{op}})), \end{equation}

where $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U} \mathfrak {g})$ is the category of $G$-equivariant finitely generated $\mathcal {U} \mathfrak {g}$-modules, and similarly for $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U} \mathfrak {g}^{\mathrm {op}})$. For instance, for the first functor, one can reconstruct the right action of $\mathcal {U} \mathfrak {g}$ on a $G$-equivariant $\mathcal {U} \mathfrak {g}$-module $M$ by setting $m \cdot x := x \cdot m -\varrho (x)(m)$ for $m \in M$ and $x \in \mathfrak {g}$, where $\varrho$ denotes the differential of the $G$-action.

We will also denote by $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}})$ the category of $G$-equivariant finitely generated (left) modules over $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$. As above, since $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$ is finitely generated both as a left and as a right $\mathcal {U} \mathfrak {g}$-module, the tensor product $\otimes _{\mathcal {U} \mathfrak {g}}$ endows this category with a monoidal structure, and as explained above we have a fully faithful monoidal functor

(3.6)\begin{equation} \mathsf{HC} \to \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U} \mathfrak{g}^{\mathrm{op}}). \end{equation}

If $M$ belongs to $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}})$, then one obtains an extra $\mathcal {U}_0 \mathfrak {g}$-action on $M$ by setting $x \cdot m = (x \otimes 1 - 1 \otimes x)m$ for $x \in \mathfrak {g}$ and $m \in M$. Since $\mathcal {U}_0 \mathfrak {g}$ identifies canonically with the distribution algebra $\mathrm {Dist}(G_1)$ of the kernel $G_1$ of $\mathrm {Fr}$, $M$ becomes in this way a $G \ltimes G_1$-equivariant $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$-module, where the action of $G \ltimes G_1$ on $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$ is obtained from the $G$-action by composition with the product morphism $G \ltimes G_1 \to G$. As for (3.5), forgetting the right (respectively, left) action of $\mathcal {U} \mathfrak {g}$ defines an equivalence of categories

\[ \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U} \mathfrak{g}^{\mathrm{op}}) \xrightarrow{\sim} \mathsf{Mod}^{G \ltimes G_1}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g}) \quad (\text{respectively, } \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U} \mathfrak{g}^{\mathrm{op}}) \xrightarrow{\sim} \mathsf{Mod}^{G \ltimes G_1}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g}^{\mathrm{op}})). \]

From the point of view of these equivalences and those in (3.5), the essential image of (3.6) consists of equivariant modules on which the action of $G \ltimes G_1$ factors through the product morphism $G \ltimes G_1 \to G$.

One can construct interesting objects in $\mathsf {HC}$ from $G$-modules as follows. Given $V$ in $\mathsf {Rep}(G)$, we consider the Harish-Chandra bimodule

\[ V \otimes \mathcal{U} \mathfrak{g}, \]

where the left $\mathcal {U} \mathfrak {g}$-action is diagonal (with respect to the action on $V$ obtained by differentiation, and the action on $\mathcal {U} \mathfrak {g}$ by left multiplication), the right $\mathcal {U} \mathfrak {g}$-action is induced by right multiplication on $\mathcal {U} \mathfrak {g}$, and the $G$-action is diagonal (with respect to the given action on $V$ and the adjoint action on $\mathcal {U} \mathfrak {g}$). In particular, for $x,y,z \in \mathcal {U} \mathfrak {g}$ and $v \in V$ we have

\[ x \cdot (v \otimes z) \cdot y = (x_{(1)} \cdot v) \otimes (x_{(2)} zy), \]

where we use Sweedler's notation for the comultiplication in the Hopf algebra $\mathcal {U} \mathfrak {g}$. It is easily seen that the map $(x \otimes y) \otimes v \mapsto (x_{(1)} \cdot v) \otimes (x_{(2)} y)$ induces an isomorphism of $G$-equivariant $\mathcal {U} \mathfrak {g}$-bimodules

\[ (\mathcal{U} \mathfrak{g} \otimes \mathcal{U} \mathfrak{g}^{\mathrm{op}}) \otimes_{\mathcal{U} \mathfrak{g}} V \xrightarrow{\sim} V \otimes \mathcal{U} \mathfrak{g}, \]

where the tensor product over $\mathcal {U} \mathfrak {g}$ on the left-hand side is taken with respect to the morphism $\mathcal {U} \mathfrak {g} \to \mathcal {U} \mathfrak {g} \otimes \mathcal {U} \mathfrak {g}^{\mathrm{op}}$ defined by $x \mapsto x_{(1)} \otimes S(x_{(2)})$, where $S$ is the antipode. In particular, the modules $V \otimes \mathcal {U} \mathfrak {g}$ are ‘induced from the diagonal’. For $V,V'$ in $\mathsf {Rep}(G)$, we have a canonical isomorphism of Harish-Chandra bimodules

(3.7)\begin{equation} (V \otimes \mathcal{U} \mathfrak{g}) \otimes_{\mathcal{U} \mathfrak{g}} (V' \otimes \mathcal{U} \mathfrak{g}) \xrightarrow{\sim} (V \otimes V') \otimes \mathcal{U} \mathfrak{g}. \end{equation}

One can similarly consider, again for $V$ in $\mathsf {Rep}(G)$, the Harish-Chandra bimodule

\[ \mathcal{U} \mathfrak{g} \otimes V \]

where now the actions of $\mathcal {U} \mathfrak {g}$ are defined by

\[ x \cdot (z \otimes v) \cdot y = (xzy_{(1)}) \otimes (S(y_{(2)}) \cdot v) \]

for $x,y,z \in \mathcal {U} \mathfrak {g}$ and $v \in V$ (and the $G$-action is still diagonal). As above, we have an isomorphism of $G$-equivariant $\mathcal {U} \mathfrak {g}$-bimodules

\[ (\mathcal{U} \mathfrak{g} \otimes \mathcal{U} \mathfrak{g}^{\mathrm{op}}) \otimes_{\mathcal{U} \mathfrak{g}} V \xrightarrow{\sim} \mathcal{U} \mathfrak{g} \otimes V, \]

now given by $(x \otimes y) \otimes v \mapsto (xy_{(1)}) \otimes (S(y_{(2)}) \cdot v)$ for $x,y \in \mathcal {U} \mathfrak {g}$ and $v \in V$. In particular, the objects $V \otimes \mathcal {U} \mathfrak {g}$ and $\mathcal {U} \mathfrak {g} \otimes V$ are isomorphic; explicitly, the isomorphism is given by

\[ v \otimes x \mapsto x \otimes (S(x) \cdot v) \]

for $x \in \mathcal {U} \mathfrak {g}$ and $v \in V$.

3.5 Completed Harish-Chandra bimodules

We now need to adapt the considerations of § 3.4 to the setting of completed Harish-Chandra characters.

Let us set

\[ \mathfrak{D} := \mathrm{Spec}({Z_{\mathrm{HC}}} \otimes_{{Z_{\mathrm{HC}}} \cap {Z_{\mathrm{Fr}}}} {Z_{\mathrm{HC}}}) \cong \mathfrak{t}^{*}/(W,\bullet) \times_{\mathfrak{t}^{*(1)}/W} \mathfrak{t}^{*}/(W,\bullet), \]

so that $\mathcal {Z} = \mathscr {O}(\mathfrak {g}^{*(1)} \times _{\mathfrak {t}^{*(1)}/W} \mathfrak {D})$. For $\lambda,\mu \in \mathbb {X}$, we will also set

\[ \mathcal{I}^{\lambda,\mu} := \mathfrak{m}^{\lambda} \otimes_{{Z_{\mathrm{HC}}} \cap {Z_{\mathrm{Fr}}}} {Z_{\mathrm{HC}}} + {Z_{\mathrm{HC}}} \otimes_{{Z_{\mathrm{HC}}} \cap {Z_{\mathrm{Fr}}}} \mathfrak{m}^{\mu} \cdot {Z_{\mathrm{HC}}}, \]

and will denote by $\mathfrak {D}^{\hat {\lambda },\hat {\mu }}$ the spectrum of the completion of $\mathscr {O}(\mathfrak {D})$ with respect to the maximal ideal $\mathcal {I}^{\lambda,\mu }$. Finally, we set

\[ \mathcal{U}^{\hat{\lambda},\hat{\mu}} := ( \mathcal{U} \mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U} \mathfrak{g}^{\mathrm{op}} ) \otimes_{\mathcal{Z}} \mathscr{O}(\mathfrak{g}^{*(1)} \times_{\mathfrak{t}^{*(1)}/W} \mathfrak{D}^{\hat{\lambda},\hat{\mu}}) \cong ( \mathcal{U} \mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U} \mathfrak{g}^{\mathrm{op}} ) \otimes_{\mathscr{O}(\mathfrak{D})} \mathscr{O}(\mathfrak{D}^{\hat{\lambda},\hat{\mu}}). \]

(Note that $\mathcal {U}^{\hat {\lambda },\hat {\mu }}$ is not the completion of $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$ with respect to the ideal generated by $\mathcal {I}^{\lambda,\mu }$, since $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$ is not of finite type as an $\mathscr {O}(\mathfrak {D})$-module.)

The algebra $\mathscr {O}(\mathfrak {D}^{\hat {\lambda },\hat {\mu }})$ is Noetherian (see [Sta20, Tag 05GH]), hence $\mathscr {O}(\mathfrak {g}^{*(1)} \times _{\mathfrak {t}^{*(1)}/W} \mathfrak {D}^{\hat {\lambda },\hat {\mu }})$ is Noetherian too, being finitely generated over $\mathscr {O}(\mathfrak {D}^{\hat {\lambda },\hat {\mu }})$ (see [Sta20, Tag 00FN]). Finally, since $\mathcal {U}^{\hat {\lambda },\hat {\mu }}$ is finitely generated as an $\mathscr {O}(\mathfrak {g}^{*(1)} \times _{\mathfrak {t}^{*(1)}/W} \mathfrak {D}^{\hat {\lambda },\hat {\mu }})$-module it is left and right Noetherian (as a non-commutative ring), and a $\mathcal {U}^{\hat {\lambda },\hat {\mu }}$-module is finitely generated if and only if it is finitely generated as an $\mathscr {O}(\mathfrak {g}^{*(1)} \times _{\mathfrak {t}^{*(1)}/W} \mathfrak {D}^{\hat {\lambda },\hat {\mu }})$-module.

The $G$-action on $\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}}$ induces an algebraic $G$-module structure on $\mathcal {U}^{\hat {\lambda },\hat {\mu }}$, and we can consider the category of $G$-equivariant finitely generated modules over this algebra; this (abelian) category will be denoted by $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\lambda },\hat {\mu }})$. The full subcategory whose objects are the modules $M$ such that the differential of the $G$-action coincides with the action given by $x \cdot m=xm-mx$ for $x \in \mathfrak {g}$ and $m \in M$ will be denoted $\mathsf {HC}^{\hat {\lambda },\hat {\mu }}$; its objects will be called completed Harish-Chandra bimodules.

Given $\lambda,\mu \in \mathbb {X}$, we have a natural exact functor

\[ \mathsf{C}^{\lambda,\mu} : \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U} \mathfrak{g}^{\mathrm{op}}) \to \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\hat{\lambda},\hat{\mu}}), \]

defined by

\[ \mathsf{C}^{\lambda,\mu}(M) = \mathscr{O}(\mathfrak{D}^{\hat{\lambda},\hat{\mu}}) \otimes_{\mathscr{O}(\mathfrak{D})} M, \]

which restricts to a functor $\mathsf {HC} \to \mathsf {HC}^{\hat {\lambda },\hat {\mu }}$. (Exactness of this functor follows from the fact that $\mathscr {O}(\mathfrak {D}^{\hat {\lambda },\hat {\mu }})$ is flat over $\mathscr {O}(\mathfrak {D})$; see [Sta20, Tag 00MB].) We will denote by

\[ \mathsf{HC}^{\hat{\lambda},\hat{\mu}}_\mathrm{diag} \]

the full additive subcategory of $\mathsf {HC}^{\hat {\lambda },\hat {\mu }}$ whose objects are direct summands of objects of the form $\mathsf {C}^{\lambda,\mu } (V \otimes \mathcal {U} \mathfrak {g})$ with $V$ in $\mathsf {Rep}(G)$. (In view of the comments at the end of § 3.4, objects in this category will sometimes be referred to as completed diagonally induced Harish-Chandra bimodules.) In the case where $\lambda =\mu$, we will set

\[ \mathcal{U}^{\hat{\lambda}} = \mathsf{C}^{\lambda,\lambda}(\Bbbk \otimes \mathcal{U} \mathfrak{g}), \]

where $\Bbbk$ is the trivial $G$-module.

For later use, we also introduce some completed bimodules which are closely related to the translation functors for $G$-modules (see § 6.3 below for details). Recall that a weight $\lambda \in \mathbb {X}$ is said to belong to the fundamental alcove (respectively, to the closure of the fundamental alcove) if it satisfies

\[ 0 < \langle \lambda+\rho, \alpha^{\vee} \rangle < p \quad \text{(respectively, } 0 \leq \langle \lambda+\rho, \alpha^{\vee} \rangle \leq p\text{)}, \]

for any positive root $\alpha$. With this notation, the set of weights which belong to the closure of the fundamental alcove is a fundamental domain for the $({W_{\mathrm {aff}}},\bullet )$-action on $\mathbb {X}$. Moreover, if $\lambda \in \mathbb {X}$ belongs to the closure of the fundamental alcove, then its stabilizer in ${W_{\mathrm {aff}}}$ is the parabolic subgroup generated by the elements $s \in {S_{\mathrm {aff}}}$ such that $s \bullet \lambda =\lambda$; see [Reference JantzenJan03, §II.6.3].

Let $\mathbb {X}^{+} \subset \mathbb {X}$ be the subset of dominant weights determined by $\mathfrak {R}^{+}$. For any $\nu \in \mathbb {X}^{+}$, we will denote by $\mathsf {L}(\nu )$ the simple $G$-module with highest weight $\nu$, that is, the unique simple submodule in $\mathrm {Ind}_{B}^{G}(\nu )$. Given two weights $\lambda,\mu \in \mathbb {X}$ which belong to the closure of the fundamental alcove, we set

\[ \mathsf{P}^{\lambda,\mu} := \mathsf{C}^{\lambda,\mu}(\mathsf{L}(\nu) \otimes \mathcal{U} \mathfrak{g}) \in \mathsf{HC}_\mathrm{diag}^{\hat{\lambda},\hat{\mu}}, \]

where $\nu$ is the unique dominant $W$-translate of $\lambda -\mu$.

3.6 Comparison of completions

For notational simplicity, let us now fix a subset $\Lambda \subset \mathbb {X}$ such that the map $\lambda \mapsto {{\tilde {\lambda }}}$ restricts to a bijection $\Lambda \xrightarrow {\sim } \mathfrak {t}^{*}_{\mathbb {F}_p}/(W,\bullet )$. (In other words, $\Lambda$ is a set of representatives for the $\bullet$-action of ${W_{\mathrm {ext}}}$ on $\mathbb {X}$.)

We will denote by $\mathcal {I} \subset \mathscr {O}(\mathfrak {t}^{*(1)}/W)={Z_{\mathrm {HC}}} \cap {Z_{\mathrm {Fr}}}$ the maximal ideal corresponding to the image of $0 \in \mathfrak {t}^{*(1)}$. Then, in the notation of § 3.2, $\mathcal {I} \cdot {Z_{\mathrm {Fr}}}$ is the ideal of definition of $\mathcal {N}^{*(1)} \subset \mathfrak {g}^{*(1)}$, and each ideal $\mathcal {I}^{\lambda,\mu }$ contains $\mathcal {I} \cdot \mathscr {O}(\mathfrak {D})$. We will denote by $\mathfrak {D}^{\wedge }$ the spectrum of the completion of $\mathscr {O}(\mathfrak {D})$ with respect to the ideal $\mathcal {I} \cdot \mathscr {O}(\mathfrak {D})$. Note that since $\mathscr {O}(\mathfrak {D})$ is finite as an $\mathscr {O}(\mathfrak {t}^{*(1)}/W)$-module (because the morphisms $\mathfrak {t}^{*} \to \mathfrak {t}^{*(1)}$ and $\mathfrak {t}^{*(1)} \to \mathfrak {t}^{*(1)}/W$ are finite), if we denote by $\mathscr {O}(\mathfrak {t}^{*(1)}/W)^{\wedge }$ the completion of $\mathscr {O}(\mathfrak {t}^{*(1)}/W)$ with respect to $\mathcal {I}$ we have a canonical isomorphism

(3.8)\begin{equation} \mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge} \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathfrak{D}) \xrightarrow{\sim} \mathscr{O}(\mathfrak{D}^{\wedge}); \end{equation}

see [Sta20, Tag 00MA].

Lemma 3.4 The natural morphism

\[ \mathscr{O}(\mathfrak{D}^{\wedge}) \to \prod_{\lambda,\mu \in \Lambda} \mathscr{O}(\mathfrak{D}^{\hat{\lambda},\hat{\mu}}) \]

is a ring isomorphism.

Proof. The lemma will basically follow from the observation that the closed points in the fiber of the morphism (3.4) over the (closed) point corresponding to $\mathcal {I}$ are those corresponding to the ideals $\mathfrak {m}^{\lambda }$ with $\lambda \in \mathbb {X}$, which itself follows from the fact that the fiber of $\mathrm {AS} : \mathfrak {t}^{*} \to \mathfrak {t}^{*(1)}$ over $0$ is $\mathfrak {t}^{*}_{\mathbb {F}_p}$.

More precisely, the morphism considered in this statement is the product of the morphisms $\mathscr {O}(\mathfrak {D}^{\wedge }) \to \mathscr {O}(\mathfrak {D}^{\hat {\lambda },\hat {\mu }})$ induced by the natural morphisms $\mathscr {O}(\mathfrak {D})/(\mathcal {I}^{n} \cdot \mathscr {O}(\mathfrak {D})) \twoheadrightarrow \mathscr {O}(\mathfrak {D})/(\mathcal {I}^{\lambda,\mu })^{n}$. This morphism is clearly a ring morphism; to prove that it is invertible we will construct its inverse.

Let us fix some $n \geq 1$, and consider the quotient $\mathscr {O}(\mathfrak {D})/(\mathcal {I}^{n} \cdot \mathscr {O}(\mathfrak {D}))$. Here, as explained above, $\mathscr {O}(\mathfrak {D})={Z_{\mathrm {HC}}} \otimes _{{Z_{\mathrm {HC}}} \cap {Z_{\mathrm {Fr}}}} {Z_{\mathrm {HC}}}$ is a finite $\mathscr {O}(\mathfrak {t}^{*(1)}/W)$-module; therefore, this algebra is finite-dimensional. Its maximal ideals are in bijection with the maximal ideals of $\mathscr {O}(\mathfrak {D})$ containing $\mathcal {I} \cdot \mathscr {O}(\mathfrak {D})$, hence with $\Lambda \times \Lambda$ through $(\lambda,\mu ) \mapsto \mathcal {I}^{\lambda,\mu } / \mathcal {I}^{n} \cdot \mathscr {O}(\mathfrak {D})$. In view of the general theory of Artin rings (see, for example, [Reference Atiyah and MacdonaldAM69, Chap. 8]), for any $\lambda,\mu \in \Lambda$ the quotient

\[ \mathscr{O}(\mathfrak{D})/(\mathcal{I}^{n} \cdot \mathscr{O}(\mathfrak{D}) + (\mathcal{I}^{\lambda,\mu})^{m}) \]

does not depend on $m$ for $m \gg 0$, and the natural morphism from $\mathscr {O}(\mathfrak {D})/(\mathcal {I}^{n} \cdot \mathscr {O}(\mathfrak {D}))$ to the product of these rings (over $\Lambda \times \Lambda$) is an isomorphism.

We are now ready to define the wished-for inverse morphism

\[ \prod_{\lambda,\mu \in \Lambda} \mathscr{O}(\mathfrak{D}^{\hat{\lambda},\hat{\mu}}) \to \mathscr{O}(\mathfrak{D}^{\wedge}). \]

For this it suffices to define, for any $n \geq 1$, a ring morphism

(3.9)\begin{equation} \prod_{\lambda,\mu \in \Lambda} \mathscr{O}(\mathfrak{D}^{\hat{\lambda},\hat{\mu}}) \to \mathscr{O}(\mathfrak{D})/(\mathcal{I}^{n} \cdot \mathscr{O}(\mathfrak{D})). \end{equation}

We fix $m$ such that the natural morphism

(3.10)\begin{equation} \mathscr{O}(\mathfrak{D})/(\mathcal{I}^{n} \cdot \mathscr{O}(\mathfrak{D})) \to \prod_{\lambda,\mu \in \Lambda} \mathscr{O}(\mathfrak{D})/(\mathcal{I}^{n} \cdot \mathscr{O}(\mathfrak{D}) + (\mathcal{I}^{\lambda,\mu})^{m}) \end{equation}

is an isomorphism. Then we have natural ring morphisms

\[ \prod_{\lambda,\mu \in \Lambda} \mathscr{O}(\mathfrak{D}^{\hat{\lambda},\hat{\mu}}) \to \prod_{\lambda,\mu \in \Lambda} \mathscr{O}(\mathfrak{D}) / (\mathcal{I}^{\lambda,\mu})^{m} \to \prod_{\lambda,\mu \in \Lambda} \mathscr{O}(\mathfrak{D})/(\mathcal{I}^{n} \cdot \mathscr{O}(\mathfrak{D}) + (\mathcal{I}^{\lambda,\mu})^{m}). \]

Composing with the inverse of (3.10), we deduce the desired map (3.9).

It is easy (and left to the reader) to check that the two morphisms considered above are inverse to each other.

Remark 3.5 If we denote by ${Z^{\wedge}_{\mathrm {HC}}}$ the completion of ${Z_{\mathrm {HC}}}$ with respect to the ideal $\mathcal {I} \cdot {Z_{\mathrm {HC}}}$, then we have as in (3.8) a canonical isomorphism $\mathscr {O}(\mathfrak {t}^{*(1)}/W)^{\wedge } \otimes _{\mathscr {O}(\mathfrak {t}^{*(1)}/W)} {Z_{\mathrm {HC}}} \xrightarrow {\sim } {Z^{\wedge }_{\mathrm {HC}}}$, and therefore a canonical isomorphism

\[ \mathscr{O}(\mathfrak{D}^{\wedge}) \xrightarrow{\sim} {Z^{\wedge}_{\mathrm{HC}}} \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge}} {Z^{\wedge}_{\mathrm{HC}}}. \]

Denoting by ${Z^{\hat {\lambda }}_{\mathrm {HC}}}$ the completion of ${Z_{\mathrm {HC}}}$ with respect to the ideal $\mathfrak {m}^{\lambda }$, for $\lambda \in \mathbb {X}$, the same considerations as for Lemma 3.4 show that we have a canonical isomorphism

\[ {Z^{\wedge}_{\mathrm{HC}}} \xrightarrow{\sim} \prod_{\lambda \in \Lambda} {Z^{\hat{\lambda}}_{\mathrm{HC}}}, \]

from which we obtain a decomposition

\[ \mathscr{O}(\mathfrak{D}^{\wedge}) \xrightarrow{\sim} \prod_{\lambda,\mu \in \Lambda} {Z^{\hat{\lambda}}_{\mathrm{HC}}} \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge}} {Z^{\hat{\mu}}_{\mathrm{HC}}}. \]

This decomposition is in fact ‘the same’ as the decomposition of Lemma 3.4, in the sense that for any $\lambda,\mu \in \Lambda$ we have an identification

(3.11)\begin{equation} \mathscr{O}(\mathfrak{D}^{\hat{\lambda},\hat{\mu}}) \xrightarrow{\sim} {Z^{\hat{\lambda}}_{\mathrm{HC}}} \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge}} {Z^{\hat{\mu}}_{\mathrm{HC}}}. \end{equation}

We will also set

\[ \mathcal{U}^{\wedge} := ( \mathcal{U}\mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U}\mathfrak{g}^{\mathrm{op}} ) \otimes_{\mathscr{O}(\mathfrak{D})} \mathscr{O}(\mathfrak{D}^{\wedge}) \cong ( \mathcal{U}\mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U}\mathfrak{g}^{\mathrm{op}} ) \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge}, \]

where the isomorphism uses (3.8). Then, as in § 3.5, $\mathcal {U}^{\wedge }$ is a left and right Noetherian ring, endowed with a structure of an algebraic $G$-module, and Lemma 3.4 implies that the natural morphism

(3.12)\begin{equation} \mathcal{U}^{\wedge} \to \prod_{\lambda,\mu \in \Lambda} \mathcal{U}^{\hat{\lambda},\hat{\mu}} \end{equation}

is an algebra isomorphism. Below we will consider various categories of $\mathcal {U}^{\wedge }$-modules; in fact we have

\begin{align*} \mathcal{U}^{\wedge} &\cong ( \mathcal{U} \mathfrak{g} \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge} ) \otimes_{{Z_{\mathrm{Fr}}} \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge}}\\ &\quad \times ( \mathcal{U} \mathfrak{g} \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge} )^{\mathrm{op}}; \end{align*}

a $\mathcal {U}^{\wedge }$-module is therefore the same as a $(\mathcal {U} \mathfrak {g} \otimes _{\mathscr {O}(\mathfrak {t}^{*(1)}/W)} \mathscr {O}(\mathfrak {t}^{*(1)}/W)^{\wedge })$-bimodule on which the left and right actions of ${Z_{\mathrm {Fr}}} \otimes _{\mathscr {O}(\mathfrak {t}^{*(1)}/W)} \mathscr {O}(\mathfrak {t}^{*(1)}/W)^{\wedge }$ coincide.

We will denote by $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\wedge })$ the abelian category of $G$-equivariant finitely generated $\mathcal {U}^{\wedge }$-modules. In view of (3.12), we have a canonical equivalence of categories

(3.13)\begin{equation} \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\wedge}) \cong \bigoplus_{\lambda,\mu \in \Lambda} \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\hat{\lambda},\hat{\mu}}). \end{equation}

We also have a canonical exact functor

(3.14)\begin{equation} \mathsf{C}^{\wedge} : \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U} \mathfrak{g}^{\mathrm{op}}) \to \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\wedge}) \end{equation}

defined by $\mathsf {C}^{\wedge }(M)=\mathscr {O}(\mathfrak {D}^{\wedge }) \otimes _{\mathscr {O}(\mathfrak {D})} M$. For the same reasons as above, for any $M$ in $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}})$ we have a canonical isomorphism

\[ \mathsf{C}^{\wedge}(M) \cong \bigoplus_{\lambda,\mu \in \Lambda} \mathsf{C}^{\lambda,\mu}(M). \]

An object $M$ in $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\wedge })$ will be called a completed Harish-Chandra bimodule if the differential of the $G$-action coincides with the action given by $x \cdot m = xm-mx$ for $x \in \mathfrak {g}$ and $m \in M$, and we will denote by $\mathsf {HC}^{\wedge }$ the full subcategory of $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\wedge })$ consisting of such objects; this terminology is compatible with that of § 3.5 in the sense that (3.13) restricts to an equivalence of categories

\[ \mathsf{HC}^{\wedge} \cong \bigoplus_{\lambda,\mu \in \Lambda} \mathsf{HC}^{\hat{\lambda},\hat{\mu}}. \]

We will denote by $\mathsf {HC}^{\wedge }_\mathrm {diag}$ the full additive subcategory of $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\wedge })$ whose objects are the direct summands of objects of the form $\mathsf {C}^{\wedge }(V \otimes \mathcal {U} \mathfrak {g})$ with $V$ in $\mathsf {Rep}(G)$. With this definition, (3.13) restricts further to an equivalence of categories

\[ \mathsf{HC}^{\wedge}_\mathrm{diag} \cong \bigoplus_{\lambda,\mu \in \Lambda} \mathsf{HC}^{\hat{\lambda},\hat{\mu}}_\mathrm{diag}. \]

3.7 Monoidal structure

We now want to define some analogue of the monoidal structure on $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}})$ for the categories $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\lambda },\hat {\mu }})$. More specifically, given $\lambda,\mu,\nu$ in $\mathbb {X}$, we want to define a canonical bifunctor

(3.15)\begin{equation} (-) {\widehat{\otimes}}_{\mathcal{U} \mathfrak{g}} (-) : \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\hat{\lambda},\hat{\mu}}) \times \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\hat{\mu},\hat{\nu}}) \to \mathsf{Mod}^{G}_{\mathrm{fg}} (\mathcal{U}^{\hat{\lambda},\hat{\nu}}) \end{equation}

right exact on both sides, which restricts to a bifunctor

\[ \mathsf{HC}^{\hat{\lambda},\hat{\mu}} \times \mathsf{HC}^{\hat{\mu},\hat{\nu}} \to \mathsf{HC}^{\hat{\lambda},\hat{\nu}}, \]

these bifunctors satisfying natural unit and associativity axioms. Explicitly, we require that:

• – if $\mu =\lambda$ we have a canonical isomorphism of functors

  \[ \mathcal{U}^{\hat{\lambda}} {\widehat{\otimes}}_{\mathcal{U} \mathfrak{g}} (-) \cong \mathrm{id}, \]
  and if $\nu =\mu$ we have a canonical isomorphism
  \[ (-) {\widehat{\otimes}}_{\mathcal{U} \mathfrak{g}} \mathcal{U}^{\hat{\mu}} \cong \mathrm{id}; \]

• – for four weights $\lambda,\mu,\nu,\eta \in \mathbb {X}$ we have an isomorphism

  \[ ( (-) {\widehat{\otimes}}_{\mathcal{U} \mathfrak{g}} (-) ) {\widehat{\otimes}}_{\mathcal{U} \mathfrak{g}} (-) \xrightarrow{\sim} (-) {\widehat{\otimes}}_{\mathcal{U} \mathfrak{g}} ( (-) {\widehat{\otimes}}_{\mathcal{U} \mathfrak{g}} (-) ) \]
  of functors from
  \[ \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\hat{\lambda},\hat{\mu}}) \times \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\hat{\mu},\hat{\nu}}) \times \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\hat{\nu},\hat{\eta}}) \]
  to $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\lambda },\hat {\eta }})$.

In particular, in the case where $\lambda =\mu =\nu$, this construction will equip $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\lambda },\hat {\lambda }})$ with the structure of a monoidal category.

For this we can assume that all the weights involved belong to the subset $\Lambda$ chosen in § 3.6. It therefore suffices to construct a monoidal structure of the category $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\wedge })$, with monoidal unit $\mathsf {C}^{\wedge }(\Bbbk \otimes \mathcal {U} \mathfrak {g})$; the bifunctor (3.15) will then be deduced by restriction to the factor $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\lambda },\hat {\mu }}) \times \mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\mu },\hat {\nu }})$ in the decomposition (3.13).

Recall that we have

\[ \mathcal{U}^{\wedge} = ( \mathcal{U}\mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U}\mathfrak{g}^{\mathrm{op}} ) \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge}. \]

Given $M,N$ in $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\wedge })$, we set

\[ M {\widehat{\otimes}}_{\mathcal{U} \mathfrak{g}} N := M \otimes_{\mathcal{U} \mathfrak{g} \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge}} N, \]

where the right action of $\mathcal {U} \mathfrak {g} \otimes _{\mathscr {O}(\mathfrak {t}^{*(1)}/W)} \mathscr {O}(\mathfrak {t}^{*(1)}/W)^{\wedge }$ on $M$ is induced by the action of the second copy of $\mathcal {U} \mathfrak {g}$, and the left action on $N$ is induced by the action of the first copy of $\mathcal {U} \mathfrak {g}$. This tensor product admits compatible actions of $\mathcal {U}\mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U}\mathfrak {g}^{\mathrm{op}}$ (induced by the action of the first copy of $\mathcal {U} \mathfrak {g}$ on $M$, and of the second copy of $\mathcal {U} \mathfrak {g}$ on $N$) and of $\mathscr {O}(\mathfrak {t}^{*(1)}/W)^{\wedge }$, hence of $\mathcal {U}^{\wedge }$. This module is, moreover, finitely generated, since it is finitely generated over $\mathscr {O}(\mathfrak {g}^{*(1)}) \otimes _{\mathscr {O}(\mathfrak {t}^{*(1)}/W)} \mathscr {O}(\mathfrak {t}^{*(1)}/W)^{\wedge }$, and it admits a canonical (diagonal) $G$-module structure. In this way we obtain a bifunctor

\[ \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\wedge}) \times \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\wedge}) \to \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\wedge}), \]

which is easily seen to provide a monoidal structure with monoidal unit $\mathsf {C}^{\wedge }(\Bbbk \otimes \mathcal {U} \mathfrak {g})$. It is easily seen also that the functor (3.14) has a canonical monoidal structure; using (3.7), we deduce that the full subcategory $\mathsf {HC}^{\wedge }_\mathrm {diag}$ is a monoidal subcategory.

If $\lambda,\mu,\nu \in \mathbb {X}$ and if $M$ belongs to $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\lambda },\hat {\mu }})$ and $N$ belongs to $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\mu },\hat {\nu }})$, then seeing $M$ and $N$ as objects in $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\wedge })$ via (3.13), the product $M {\widehat{\otimes}}_{\mathcal {U} \mathfrak {g}} N$ belongs to the factor $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\lambda },\hat {\nu }})$, which provides the desired bifunctor (3.15). (In fact, the action of the left copy of ${Z_{\mathrm {HC}}}$ factors through an action of ${Z^{\hat {\lambda }}_{\mathrm {HC}}}$, and that of the right copy factors through an action of ${Z^{\hat {\nu }}_{\mathrm {HC}}}$, which justifies the claim in view of (3.11).) From the corresponding property for $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\wedge })$ we deduce that the subcategories $\mathsf {HC}^{\hat {\lambda },\hat {\mu }}$ and $\mathsf {HC}^{\hat {\lambda },\hat {\mu }}_\mathrm {diag}$ are stable (in the obvious sense) under this bifunctor. In this setting the functor $\mathsf {C}^{\hat {\lambda },\hat {\mu }}$ satisfies

\[ \mathsf{C}^{\hat{\lambda},\hat{\mu}}(M \otimes_{\mathcal{U} \mathfrak{g}} N) \cong \bigoplus_{\nu \in \Lambda} \mathsf{C}^{\hat{\lambda},\hat{\nu}}(M) {\widehat{\otimes}}_{\mathcal{U} \mathfrak{g}} \mathsf{C}^{\hat{\nu},\hat{\mu}}(N) \]

for any $M,N$ in $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U} \mathfrak {g} \otimes _{{Z_{\mathrm {Fr}}}} \mathcal {U} \mathfrak {g}^{\mathrm {op}})$.

3.8 Restriction to the Kostant section

Recall the constructions of § 2.3 applied to the group $\mathbf {G}=G^{(1)}$. In particular, we have a Kostant section $\mathbf {S}^{*} \subset \mathbf {g}^{*}=\mathfrak {g}^{*(1)}$, and group schemes $\mathbb {J}^{*}_\mathrm {reg}$ over $\mathfrak {g}_\mathrm {reg}^{*(1)}$ and $\mathbb {J}^{*}_\mathbf {S}$ over $\mathbf {S}^{*}$.

We also set

\[ \mathbb{I}^{*}_\mathbf{S} := (G \times \mathbf{S}^{*}) \times_{G^{(1)} \times \mathbf{S}^{*}} \mathbb{J}^{*}_\mathbf{S}, \]

where the map $G \times \mathbf {S}^{*} \to G^{(1)} \times \mathbf {S}^{*}$ is the product of the Frobenius morphism of $G$ and the identity of $\mathbf {S}^{*}$. Since $G$ is smooth its Frobenius morphism is flat (see, for example, [Reference Brion and KumarBK05, § 1.1]), and therefore $\mathbb {I}^{*}_\mathbf {S}$ is a flat affine group scheme over $\mathbf {S}^{*}$. By construction $\mathbb {I}^{*}_\mathbf {S}$ contains $G_1 \times \mathbf {S}^{*}$ as a normal subgroup, and the quotient identifies with $\mathbb {J}^{*}_\mathbf {S}$. Note also that the morphism (2.4) induces (after restriction to $\mathbf {S}$ and composition with the projection $\mathbb {I}^{*}_\mathbf {S} \to \mathbb {J}^{*}_\mathbf {S}$) a group-scheme morphism

(3.16)\begin{equation} \mathfrak{t}^{*(1)} \times_{\mathfrak{t}^{*(1)}/W} \mathbb{I}^{*}_\mathbf{S} \to (\mathfrak{t}^{*(1)} \times_{\mathfrak{t}^{*(1)}/W} \mathbf{S}^{*}) \times T^{(1)}. \end{equation}

Finally, we set

\[ \mathcal{U}_{\mathbf{S}}\mathfrak{g} := \mathcal{U} \mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathscr{O}(\mathbf{S}^{*}), \]

where $\mathscr {O}(\mathbf {S}^{*})$ is seen as a ${Z_{\mathrm {Fr}}}$-algebra via the identification (3.2). If we set

\[ \mathfrak{C}_\mathbf{S} := \mathbf{S}^{*} \times_{\mathfrak{t}^{*(1)} / W} \mathfrak{t}^{*}/(W,\bullet), \]

then the projection $\mathfrak {C}_\mathbf {S} \to \mathfrak {t}^{*}/(W,\bullet )$ is an isomorphism, and $\mathcal {U}_{\mathbf {S}}\mathfrak {g}$ is an $\mathscr {O}(\mathfrak {C}_\mathbf {S})$-algebra. Recall that the algebra $\mathcal {U}\mathfrak {g}$ can be seen as a $G$-equivariant $\mathscr {O}(\mathfrak {C})$-algebra (see § 3.2). Using the general construction recalled in [Reference Mautner and RicheMR18, § 2.2], from this we deduce on $\mathcal {U}_{\mathbf {S}} \mathfrak {g}$ a natural module structure for the flat affine group scheme $\mathfrak {C}_\mathbf {S} \times _{\mathbf {S}^{*}} \mathbb {I}^{*}_\mathbf {S}$ over $\mathfrak {C}_\mathbf {S}$, such that the multiplication morphism is equivariant.

3.9 (Completed) Harish-Chandra bimodules for $\mathcal {U}_{\mathbf {S}}\mathfrak {g}$

We now want to define, given $\lambda,\mu \in \mathbb {X}$, categories analogous to $\mathsf {Mod}^{G}_{\mathrm {fg}}(\mathcal {U}^{\hat {\lambda },\hat {\mu }})$ and $\mathsf {HC}^{\hat {\lambda },\hat {\mu }}$ but for the algebra $\mathcal {U}_{\mathbf {S}}\mathfrak {g}$ in place of $\mathcal {U} \mathfrak {g}$. We start with the non-completed version.

Let us consider the category $\mathsf {Mod}^{\mathbb {I}}_{\mathrm {fg}}(\mathcal {U}_\mathbf {S} \mathfrak {g} \otimes _{\mathscr {O}(\mathbf {S}^{*})} \mathcal {U}_\mathbf {S} \mathfrak {g}^{\mathrm {op}})$ of finitely generated $\mathcal {U}_\mathbf {S} \mathfrak {g} \otimes _{\mathscr {O}(\mathbf {S}^{*})} \mathcal {U}_\mathbf {S} \mathfrak {g}^{\mathrm {op}}$-modules endowed with a compatible $\mathbb {I}^{*}_\mathbf {S}$-module structure. Since $\mathbb {I}^{*}_\mathbf {S}$ is flat over $\mathbf {S}^{*}$, this category is abelian. Here $\mathcal {U}_\mathbf {S} \mathfrak {g} \otimes _{\mathscr {O}(\mathbf {S}^{*})} (\mathcal {U}_\mathbf {S} \mathfrak {g})^{\mathrm {op}}$ is an algebra over

\[ \mathcal{Z}_\mathbf{S} := \mathscr{O}(\mathfrak{t}^{*}/(W,\bullet)) \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathbf{S}^{*}) \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathfrak{t}^{*}/(W,\bullet)), \]

which identifies with $\mathscr {O}(\mathfrak {D})$ via the composition of natural algebra morphisms

\[ \mathscr{O}(\mathfrak{D}) \to \mathscr{O}(\mathfrak{g}^{*(1)} \times_{\mathfrak{t}^{*(1)}/W} \mathfrak{D}) = \mathcal{Z} \to \mathcal{Z}_\mathbf{S}. \]

As in § 3.4, the tensor product $\otimes _{\mathcal {U}_\mathbf {S} \mathfrak {g}}$ defines a monoidal structure on this category, and, using the construction of [Reference Mautner and RicheMR18, § 2.2] considered above, the functor $\mathscr {O}(\mathbf {S}^{*}) \otimes _{{Z_{\mathrm {Fr}}}} (-)$ defines a monoidal functor

(3.17)\begin{equation} \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U} \mathfrak{g} \otimes_{{Z_{\mathrm{Fr}}}} \mathcal{U} \mathfrak{g}^{\mathrm{op}}) \to \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S} \mathfrak{g} \otimes_{\mathscr{O}(\mathbf{S}^{*})} \mathcal{U}_\mathbf{S} \mathfrak{g}^{\mathrm{op}}). \end{equation}

An object $M$ in $\mathsf {Mod}^{\mathbb {I}}_{\mathrm {fg}}(\mathcal {U}_\mathbf {S} \mathfrak {g} \otimes _{\mathscr {O}(\mathbf {S}^{*})} \mathcal {U}_\mathbf {S} \mathfrak {g}^{\mathrm {op}})$ will be called a Harish-Chandra $\mathcal {U}_\mathbf {S} \mathfrak {g}$-bimodule if the restriction of the action of $\mathbb {I}^{*}_\mathbf {S}$ to $G_1 \times \mathbf {S}^{*}$, seen as an action of the algebra $\mathrm {Dist}(G_1) = \mathcal {U}_0 \mathfrak {g}$, coincides with the action determined by the rule $x \cdot m = xm-mx$ for $x \in \mathfrak {g}$ and $m \in M$. We will denote by $\mathsf {HC}_\mathbf {S}$ the full subcategory of $\mathsf {Mod}^{\mathbb {I}}_{\mathrm {fg}}(\mathcal {U}_\mathbf {S} \mathfrak {g} \otimes _{\mathscr {O}(\mathbf {S}^{*})} \mathcal {U}_\mathbf {S} \mathfrak {g}^{\mathrm {op}})$ consisting of such objects; then the functor (3.17) restricts to a functor

\[ \mathsf{HC} \to \mathsf{HC}_{\mathbf{S}}. \]

As in § 3.4, for any $M$ in $\mathsf {Mod}^{\mathbb {I}}_{\mathrm {fg}}(\mathcal {U}_\mathbf {S} \mathfrak {g} \otimes _{\mathscr {O}(\mathbf {S}^{*})} \mathcal {U}_\mathbf {S} \mathfrak {g}^{\mathrm {op}})$ the action of $\mathbb {I}^{*}_\mathbf {S}$ on $M$ extends to an action of the semi-direct product

\[ \mathbb{I}^{*}_\mathbf{S} \ltimes G_1 = \mathbb{I}^{*}_\mathbf{S} \ltimes_\mathbf{S} (G_1 \times \mathbf{S}); \]

the Harish-Chandra $\mathcal {U}_\mathbf {S} \mathfrak {g}$-bimodules are those objects on which this action factors through the multiplication morphism $\mathbb {I}^{*}_\mathbf {S} \ltimes G_1 \to \mathbb {I}^{*}_\mathbf {S}$.

Now we add completions to the picture. Given $\lambda,\mu \in \mathbb {X}$, we will denote by $\mathcal {Z}_\mathbf {S}^{\hat {\lambda },\hat {\mu }}$ the completion of $\mathcal {Z}_\mathbf {S}$ with respect to the maximal ideal $\mathcal {I}^{\lambda,\mu }_\mathbf {S} := \mathcal {I}^{\lambda,\mu } \cdot \mathcal {Z}_\mathbf {S}$, so that we have a canonical isomorphism $\mathscr {O}(\mathfrak {D}^{\hat {\lambda },\hat {\mu }}) \xrightarrow {\sim } \mathcal {Z}_\mathbf {S}^{\hat {\lambda },\hat {\mu }}$. We also set

\[ \mathcal{U}^{\hat{\lambda},\hat{\mu}}_\mathbf{S} := \mathcal{Z}_\mathbf{S}^{\hat{\lambda},\hat{\mu}} \otimes_{\mathcal{Z}_\mathbf{S}} \bigl( \mathcal{U}_\mathbf{S} \mathfrak{g} \otimes_{\mathscr{O}(\mathbf{S}^{*})} \mathcal{U}_\mathbf{S} \mathfrak{g}^{\mathrm{op}} \bigr). \]

In this setting $\mathcal {U}_\mathbf {S} \mathfrak {g} \otimes _{\mathscr {O}(\mathbf {S}^{*})} \mathcal {U}_\mathbf {S} \mathfrak {g}^{\mathrm{op}}$ is finitely generated as a $\mathcal {Z}_\mathbf {S}$-module, so that $\mathcal {U}^{\hat {\lambda },\hat {\mu }}_\mathbf {S}$ identifies with the completion of $\mathcal {U}_\mathbf {S} \mathfrak {g} \otimes _{\mathscr {O}(\mathbf {S}^{*})} \mathcal {U}_\mathbf {S} \mathfrak {g}^{\mathrm{op}}$ with respect to the ideal $\mathcal {I}^{\lambda,\mu } \cdot (\mathcal {U}_\mathbf {S} \mathfrak {g} \otimes _{\mathscr {O}(\mathbf {S}^{*})} \mathcal {U}_\mathbf {S} \mathfrak {g}^{\mathrm{op}})$. If we denote by $\mathbb {I}^{\hat {\lambda },\hat {\mu }}_\mathbf {S}$ the pullback of $\mathbb {I}^{*}_\mathbf {S}$ under the natural morphism $\mathrm {Spec}(\mathcal {Z}_\mathbf {S}^{\hat {\lambda },\hat {\mu }}) \to \mathbf {S}^{*}$, then $\mathbb {I}^{\hat {\lambda },\hat {\mu }}_\mathbf {S}$ is a flat group scheme over $\mathrm {Spec}(\mathcal {Z}_\mathbf {S}^{\hat {\lambda },\hat {\mu }})$, and the $\mathbb {I}^{*}_\mathbf {S}$-module structure on $\mathcal {U}_\mathbf {S} \mathfrak {g}$ induces a natural $\mathbb {I}^{\hat {\lambda },\hat {\mu }}_\mathbf {S}$-module structure on $\mathcal {U}^{\hat {\lambda },\hat {\mu }}_\mathbf {S}$.

The algebra $\mathcal {U}^{\hat {\lambda },\hat {\mu }}_\mathbf {S}$ is left and right Noetherian, and we will denote by $\mathsf {Mod}^{\mathbb {I}}_{\mathrm {fg}}(\mathcal {U}_\mathbf {S}^{\hat {\lambda },\hat {\mu }})$ the abelian category of $\mathbb {I}^{\hat {\lambda },\hat {\mu }}_\mathbf {S}$-equivariant finitely generated $\mathcal {U}_\mathbf {S}^{\hat {\lambda },\hat {\mu }}$-modules. Note that we have

\[ \mathcal{U}^{\hat{\lambda},\hat{\mu}}_\mathbf{S} = \mathcal{U}^{\hat{\lambda},\hat{\mu}} \otimes_{{Z_{\mathrm{Fr}}}} \mathscr{O}(\mathbf{S}^{*}); \]

in particular, the functor $\mathscr {O}(\mathbf {S}^{*}) \otimes _{{Z_{\mathrm {Fr}}}} (-)$ defines a natural functor

(3.18)\begin{equation} \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\hat{\lambda},\hat{\mu}}) \to \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\lambda},\hat{\mu}}). \end{equation}

(Standard arguments show that this functor is exact, but this will not play any role below.) If $\lambda,\mu \in \mathbb {X}$ belong to the closure of the fundamental alcove, we will also set

\[ \mathsf{P}^{\lambda,\mu}_\mathbf{S} := \mathscr{O}(\mathbf{S}^{*}) \otimes_{\mathscr{O}(\mathfrak{g}^{*(1)})} \mathsf{P}^{\lambda,\mu}. \]

One defines the notion of completed Harish-Chandra $\mathcal {U}_\mathbf {S} \mathfrak {g}$-bimodules by imposing the same condition as for $\mathsf {HC}_\mathbf {S}$. The full subcategory of $\mathsf {Mod}^{\mathbb {I}}_{\mathrm {fg}}(\mathcal {U}_\mathbf {S}^{\hat {\lambda },\hat {\mu }})$ consisting of such objects will be denoted by $\mathsf {HC}_\mathbf {S}^{\hat {\lambda },\hat {\mu }}$; it is clear that the functor (3.18) restricts to a functor

\[ \mathsf{HC}^{\hat{\lambda},\hat{\mu}} \to \mathsf{HC}_\mathbf{S}^{\hat{\lambda},\hat{\mu}}. \]

Using considerations similar to those of § 3.7, one constructs, again for $\lambda,\mu,\nu \in \mathbb {X}$, a canonical bifunctor

(3.19)\begin{equation} (-) {\widehat{\otimes}}_{\mathcal{U}_\mathbf{S} \mathfrak{g}} (-) : \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\lambda},\hat{\mu}}) \times \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\mu},\hat{\nu}}) \to \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\lambda},\hat{\nu}}) \end{equation}

which factors through a bifunctor

\[ \mathsf{HC}_\mathbf{S}^{\hat{\lambda},\hat{\mu}} \times \mathsf{HC}_\mathbf{S}^{\hat{\mu},\hat{\nu}} \to \mathsf{HC}_\mathbf{S}^{\hat{\lambda},\hat{\nu}}, \]

this construction being unital, associative and compatible in the natural way with the bifunctors $(-) {\widehat{\otimes}}_{\mathcal {U} \mathfrak {g}} (-)$ via the functors (3.18). More explicitly, one remarks that if $\mathcal {Z}_\mathbf {S}^{\wedge }$ is the completion of $\mathcal {Z}_\mathbf {S}$ with respect to the ideal $\mathcal {I} \cdot \mathcal {Z}_\mathbf {S}$, then if we set

\[ \mathcal{U}_\mathbf{S}^{\wedge} := \mathcal{Z}_\mathbf{S}^{\wedge} \otimes_{\mathcal{Z}_\mathbf{S}} ( \mathcal{U}_\mathbf{S} \mathfrak{g} \otimes_{\mathscr{O}(\mathbf{S}^{*})} \mathcal{U}_\mathbf{S} \mathfrak{g}^{\mathrm{op}} ) = \mathscr{O}(\mathbf{S}^{*(1)}) \otimes_{\mathscr{O}(\mathfrak{g}^{*(1)})} \mathcal{U}^{\wedge}, \]

as in (3.12) we have a canonical algebra isomorphism

\[ \mathcal{U}_\mathbf{S}^{\wedge} \xrightarrow{\sim} \prod_{\lambda,\mu \in \Lambda} \mathcal{U}_\mathbf{S}^{\hat{\lambda},\hat{\mu}}. \]

If we denote by $\mathbb {I}_\mathbf {S}^{\wedge }$ the pullback of $\mathbb {I}^{*}_\mathbf {S}$ to $\mathrm {Spec}(\mathcal {Z}_\mathbf {S}^{\wedge })$, then we can consider the abelian category $\mathsf {Mod}_{\mathrm {fg}}^{\mathbb {I}}(\mathcal {U}_\mathbf {S}^{\wedge })$ of finitely generated $\mathbb {I}_\mathbf {S}^{\wedge }$-equivariant $\mathcal {U}_\mathbf {S}^{\wedge }$-modules, and the bifunctor

(3.20)\begin{equation} (-) {\widehat{\otimes}}_{\mathcal{U}_\mathbf{S} \mathfrak{g}} (-) : \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}) \times \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}) \to \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}) \end{equation}

defined by

\[ M {\widehat{\otimes}}_{\mathcal{U}_\mathbf{S} \mathfrak{g}} N = M \otimes_{\mathcal{U}_\mathbf{S} \mathfrak{g} \otimes_{\mathscr{O}(\mathfrak{t}^{*(1)}/W)} \mathscr{O}(\mathfrak{t}^{*(1)}/W)^{\wedge}} N \]

defines a monoidal structure on this category. Note that any finitely generated $\mathcal {U}_\mathbf {S}^{\wedge }$-module $M$ is also finitely generated as a $\mathcal {Z}_\mathbf {S}^{\wedge }$-module, so that the natural morphism $M \to \varprojlim _n M/\mathcal {I}^{n} \cdot M$ is an isomorphism (see [Sta20, Tag 00MA]); the monoidal product considered above therefore satisfies

\[ M {\widehat{\otimes}}_{\mathcal{U}_\mathbf{S} \mathfrak{g}} N \cong \varprojlim_{n \geq 1} \, (M/\mathcal{I}^{n} \cdot M) \otimes_{\mathcal{U}_\mathbf{S} \mathfrak{g}} (N / \mathcal{I}^{n} \cdot N) \]

as $\mathcal {U}_\mathbf {S}^{\wedge }$-modules, for any $M,N$ in $\mathsf {Mod}_{\mathrm {fg}}^{\mathbb {I}}(\mathcal {U}_\mathbf {S}^{\wedge })$. The functor $\mathscr {O}(\mathbf {S}^{*}) \otimes _{{Z_{\mathrm {Fr}}}} (-)$ also induces a functor

(3.21)\begin{equation} \mathsf{Mod}^{G}_{\mathrm{fg}}(\mathcal{U}^{\wedge}) \to \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}) \end{equation}

which admits a canonical monoidal structure. The composition of this functor with the functor $\mathsf {C}^{\wedge }$ of (3.14) will be denoted by $\mathsf {C}^{\wedge }_\mathbf {S}$.

As in (3.13), we have

(3.22)\begin{equation} \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\wedge}) \cong \bigoplus_{\lambda,\mu \in \Lambda} \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\lambda},\hat{\mu}}), \end{equation}

and the bifunctor (3.19) is then obtained by restriction of (3.20) to the appropriate summands. In the case where $\lambda =\mu =\nu$, this bifunctor equips $\mathsf {Mod}^{\mathbb {I}}_{\mathrm {fg}}(\mathcal {U}_\mathbf {S}^{\hat {\lambda },\hat {\lambda }})$ with a monoidal category structure, with unit object

\[ \mathcal{U}_\mathbf{S}^{\hat{\lambda}} := \mathscr{O}(\mathbf{S}^{*}) \otimes_{\mathscr{O}(\mathfrak{g}^{*(1)})} \mathcal{U}^{\hat{\lambda}}. \]

It is clear that the functor (3.18) is compatible with the bifunctors $(-) {\widehat{\otimes}}_{\mathcal {U} \mathfrak {g}} (-)$ and $(-) {\widehat{\otimes}}_{\mathcal {U}_\mathbf {S} \mathfrak {g}} (-)$ in the obvious way.

Lemma 3.6 For any $\lambda,\mu \in \mathbb {X}$ which belong to the closure of the fundamental alcove and any $\nu \in \mathbb {X}$, the functor

\[ \mathsf{P}^{\lambda,\mu}_\mathbf{S} {\widehat{\otimes}}_{\mathcal{U}_\mathbf{S} \mathfrak{g}} (-) : \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\mu},\hat{\nu}}) \to \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\lambda},\hat{\nu}}) \]

is both left and right adjoint to the functor

\[ \mathsf{P}^{\mu,\lambda}_\mathbf{S} {\widehat{\otimes}}_{\mathcal{U}_\mathbf{S} \mathfrak{g}} (-) : \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\lambda},\hat{\nu}}) \to \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\mu},\hat{\nu}}). \]

A similar property holds for the functors $(-) {\widehat{\otimes}}_{\mathcal {U}_\mathbf {S} \mathfrak {g}} \mathsf {P}^{\lambda,\mu }_\mathbf {S}$ and $(-) {\widehat{\otimes}}_{\mathcal {U}_\mathbf {S} \mathfrak {g}} \mathsf {P}^{\mu,\lambda }_\mathbf {S}$.

Proof. We prove the case of convolution on the left; convolution on the right can be treated similarly. We remark that for any $V \in \mathsf {Rep}(G)$, the functor

\[ \mathsf{C}_\mathbf{S}^{\wedge}(V \otimes \mathcal{U} \mathfrak{g}) {\widehat{\otimes}}_{\mathcal{U}_\mathbf{S} \mathfrak{g}} (-) : \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}) \to \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}) \]

is both left and right adjoint to the functor

\[ \mathsf{C}_\mathbf{S}^{\wedge}(V^{*} \otimes \mathcal{U} \mathfrak{g}) {\widehat{\otimes}}_{\mathcal{U}_\mathbf{S} \mathfrak{g}} (-) : \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}) \to \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}). \]

(In fact, these functors can be realized more concretely as tensor product with $V$ and $V^{*}$ respectively.) On the other hand, the inclusion functor

\[ \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\lambda},\hat{\nu}}) \to \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}) \]

(see (3.22)) is both left and right adjoint to the corresponding projection functor

\[ \mathsf{Mod}_{\mathrm{fg}}^{\mathbb{I}}(\mathcal{U}_\mathbf{S}^{\wedge}) \to \mathsf{Mod}^{\mathbb{I}}_{\mathrm{fg}}(\mathcal{U}_\mathbf{S}^{\hat{\lambda},\hat{\nu}}), \]

and similarly for $\mu$ in place of $\lambda$. The desired claim follows, since the functors $\mathsf {P}^{\lambda,\mu }_\mathbf {S} {\widehat{\otimes}}_{\mathcal {U}_\mathbf {S} \mathfrak {g}} (-)$ and $\mathsf {P}^{\mu,\lambda }_\mathbf {S} {\widehat{\otimes}}_{\mathcal {U}_\mathbf {S} \mathfrak {g}} (-)$ are isomorphic to compositions of functors of this form. (More specifically, if $\nu \in \mathbb {X}$ is the only dominant $W$-translate of $\mu -\lambda$, the functor $\mathsf {P}^{\lambda,\mu }_\mathbf {S} {\widehat{\otimes}}_{\mathcal {U}_\mathbf {S} \mathfrak {g}} (-)$ involves the module $\mathsf {L}(\nu )$, and the functor $\mathsf {P}^{\mu,\lambda }_\mathbf {S} {\widehat{\otimes}}_{\mathcal {U}_\mathbf {S} \mathfrak {g}} (-)$ involves the module $\mathsf {L}(-w_0(\nu ))$; here we fix an isomorphism $\mathsf {L}(\nu )^{*} \cong \mathsf {L}(-w_0(\nu ))$.)

3.10 Restriction to the Kostant section for diagonally induced bimodules

In this subsection we aim to prove the following claim.

The proof of this proposition will use some standard properties stated in the following lemma. Here, $k$ is a commutative ring, $A$ is a left Noetherian $k$-algebra, and $H$ is an affine $k$-group scheme. For any commutative $k$-algebra $k'$ we set $H_{k'}:=\mathrm {Spec}(k') \times _{\mathrm {Spec}(k)} H$. If $A$ admits an action of $H$ by algebra automorphisms, we denote by $\mathsf {Mod}^{H}(A)$ the category of $H$-equivariant $A$-modules. (This category is abelian if $H$ is flat over $k$.)