2.1 Notation

By

\[ \omega_n=\frac{\pi^{{n}/{2}}}{\Gamma({n}/{2}+1)} \]

we denote the volume of the $n$-dimensional unit ball. The space of smooth complex valued $k$-forms on a manifold is denoted by $\Omega ^{k}(M)$. The space of smooth complex valued measures on $M$ is $\mathcal {M}^{\infty }(M)$. The space of generalized measures, also called distributions, is denoted by

\[ \mathcal{M}^{-\infty}(M):=(C^{\infty}_c(M))^{*}, \]

where here and in the following the subscript $c$ denotes compactly supported objects. Similarly, for $m=\dim M$, we denote the space of $k$-dimensional currents on $M$ by

\[ \Omega_{-\infty}^{m-k}(M):=(\Omega^{k}_c(M))^{*}. \]

The elements of this space can also be thought of as generalized $(m-k)$-forms.

For an oriented $k$-dimensional submanifold $X \subset M$, we let $[\kern-1pt[ X]\kern-1pt]$ be the $k$-current which is integration over $X$.

By $\mathbb {P}_M:=\mathbb {P}_+(T^{*}M)$ we denote the cosphere bundle of $M$, which consists of all pairs $(p,[\xi ])$, $p \in M$, $\xi \in T_p^{*}M \setminus 0$, where $[\xi ]=[\xi ']$ if there is some $\lambda >0$ with $\xi =\lambda \xi '$. When no confusion can arise, we use the same notation for subsets of $\mathbb {P}_M$ and their lifts to $T^{*}M$. The natural involution on $\mathbb {P}_M$ is the fiberwise antipodal map $s(p,[\xi ]):=(p,[-\xi ])$.

The wave front set of a generalized form $\omega \in \Omega _{-\infty }(M)$ is a closed subset of $\mathbb {P}_M$, denoted by $\operatorname {WF}(\omega )$, and we refer the reader to [Reference HörmanderHör03] or [Reference DuistermaatDui11] for details.

For a generalized pseudo-sphere $M$, we denote by $\operatorname {Geod}_k(M)$ the space of totally geodesic $k$-dimensional submanifolds of $M$.

2.2 Smooth valuations

Let $M$ be a smooth manifold of dimension $m$, which we assume oriented for simplicity.

Let $\mathcal {P}(M)$ be the set of compact differentiable polyhedra on $M$. To $A\in \mathcal {P}(M)$ we associate two subsets of $\mathbb {P}_M$. The conormal cycle, denoted $\operatorname {nc}(A)$, is the union of all conormal cones to $A$. It is an oriented closed Lipschitz submanifold of dimension $(m-1)$, and naturally stratified by locally closed smooth submanifolds corresponding to the strata of $A$.

The conormal bundle, denoted $N^{*}A$, is the union of the conormal bundles to all smooth strata of $A$. It holds that $\operatorname {nc}(A) \subset N^{*}A$. By definition, two stratified spaces intersect transversally if all pairs of smooth strata are transversal.

A smooth valuation is a functional $\mu :\mathcal {P}(M)\to \mathbb {R}$ of the form

\[ \mu(A)=\int_A \phi+\int_{\operatorname{nc}(A)} \omega, \quad \phi \in \Omega^{m}(M), \omega \in \Omega^{m-1}(\mathbb{P}_M). \]

We write $\mu =[[\phi,\omega ]]$ in this case.

The space of smooth valuations is denoted by $\mathcal {V}^{\infty }(M)$. It admits a natural filtration

\[ \mathcal{V}^{\infty}(M)=\mathcal{W}_0^{\infty}(M) \supset \mathcal{W}_1^{\infty}(M) \supset \cdots \supset \mathcal{W}_m^{\infty}(M) =\mathcal{M}^{\infty}(M). \]

It is compatible with the Alesker product of valuations.

2.3 Generalized valuations

The space of generalized valuations is

\[ \mathcal{V}^{-\infty}(M):=(\mathcal{V}^{\infty}_c(M))^{*}. \]

By Alesker–Poincaré duality we have a natural embedding $\mathcal {V}^{\infty }(M) \hookrightarrow \mathcal {V}^{-\infty }(M)$.

There is a natural filtration

\[ \mathcal{V}^{-\infty}(M)=\mathcal{W}_0^{-\infty}(M) \supset \mathcal{W}_1^{-\infty}(M) \supset \cdots \supset \mathcal{W}_m^{-\infty}(M) =\mathcal{M}^{-\infty}(M). \]

In particular, we may consider a generalized measure as a generalized valuation.

A compact differentiable polyhedron $A$ defines a generalized valuation $\chi _A$ by

\[ \langle \chi_A,\mu\rangle=\mu(A), \quad \mu \in \mathcal{V}_c^{\infty}(M). \]

A generalized valuation $\psi$ can be represented by two generalized forms $\zeta \in C^{-\infty }(M)$, $\tau \in \Omega _{-\infty }^{m}(\mathbb {P}_M)$ such that

\[ \langle \psi,[[\phi,\omega]]\rangle=\langle \zeta,\phi\rangle+\langle \tau,\omega\rangle. \]

We refer to $\zeta$ and $\tau$ as the defining currents. For instance, the defining currents of $\chi _A$, $A \in \mathcal {P}(M)$, are $\zeta =\mathbb {1}_A, \tau =[\kern-1pt[ \operatorname {nc}(A) ]\kern-1pt]$. The wave front set of $\psi$ is defined as the pair $\Lambda \subset \mathbb {P}_M$, $\Gamma \subset \mathbb {P}_{\mathbb {P}_M}$ of the wave front sets of $\zeta$ and $\tau$. The space of all generalized valuations with wave front sets contained in $\Lambda,\Gamma$ is denoted by $\mathcal {V}^{-\infty }_{\Lambda,\Gamma }(M)$.

Consider $\psi \in \mathcal {V}^{-\infty }(M)$. We say that $A \in \mathcal {P}(M)$ is WF-transversal to $\psi$, denoted $A \pitchfork \psi$, if the conditions of [Reference Alesker and BernigAB12, Theorem 8.3] hold for

\begin{align*} (\Lambda_1, \Gamma_1) & :=\operatorname{WF}(\chi_A),\\ (\Lambda_2,\Gamma_2) & :=\operatorname{WF}(\psi). \end{align*}

These conditions imply that Alesker's product of smooth valuations can be extended to a jointly sequentially continuous product

\[ \mathcal{V}_{\Lambda_1,\Gamma_1}^{-\infty}(M) \times \mathcal{V}_{\Lambda_2,\Gamma_2}^{-\infty}(M) \to \mathcal{V}^{-\infty}(M), \]

and in particular the pairing $\psi (A):=\langle \psi, \chi _A\rangle =\int _M \psi \cdot \chi _A$ is well-defined.

Let us write a sufficient set of conditions in a particular case.

2.4 Intrinsic volumes on pseudo-Riemannian manifolds

In [Reference Bernig, Faifman and SolanesBFS22] we constructed a sequence of complex-valued generalized valuations $\mu _0^{M},\ldots,\mu _m^{M}$ naturally associated to a pseudo-Riemannian manifold $M$ of dimension $m$ and extending the Lipschitz–Killing curvatures therein. They are invariant under isometries and called the intrinsic volumes of $M$. The intrinsic volume $\mu _0$ equals the Euler characteristic, whereas $\mu _m$ is the volume measure of $M$, multiplied by $\mathbf {i}^{q}$ where $q$ is the negative index of the signature. For other values of $k$, $\mu _k$ is typically neither real nor purely imaginary.

The wave front set of $\mu _k$ is contained in $(\emptyset,N^{*}(\operatorname {LC}^{*}_M))$, where $\operatorname {LC}^{*}_M \subset \mathbb {P}_M$ is the dual light cone of the metric, i.e. the set of all pairs $(p,[\xi ]) \in \mathbb {P}_M$ such that $g|_p(\xi,\xi )=0$. Here we use the metric to identify $TM$ and $T^{*}M$.

A subset $A\in \mathcal {P}(M)$ is LC-transversal if $\operatorname {nc}(A)\pitchfork \operatorname {LC}^{*}_M$.

We also need the notion of LC-regularity, which was introduced in [Reference Bernig, Faifman and SolanesBFS22].

It was shown in [Reference Bernig, Faifman and SolanesBFS22, Proposition 4.9] that the extrinsic notion of LC-transversality and the intrinsic notion of LC-regularity coincide: a submanifold of a pseudo-Riemannian manifold, equipped with the field of quadratic forms induced from the metric, is LC-regular if and only if it is LC-transversal.

The most important property of the intrinsic volumes is that they satisfy a Weyl principle: for any isometric immersion $M \looparrowright \widetilde M$ of pseudo-Riemannian manifolds we have

\[ \mu_k^{\widetilde M}|_M=\mu_k^{M}, \]

in particular, the restriction on the left-hand side is well-defined. Conversely, we have shown in [Reference Bernig, Faifman and SolanesBFS21] that any family of valuations associated to pseudo-Riemannian manifolds that satisfies the Weyl principle must be a linear combination of intrinsic volumes.

3.1 The general setting

For a submanifold with corners $X\subset M$, define $Z_X\subset X\times \operatorname {Gr}_{d-k}(V)$ by $Z_X=\{(x,E): x\in X\cap E\}$. Then $Z_X$ is a manifold with corners, more precisely it is the total space of the fiber bundle over $X$ with fiber $\operatorname {Gr}_{d-k-1}(V/\mathbb {R} x)$ at $x\in X$. Write

for the natural projections.

Denote by $W_X\subset \operatorname {Gr}_{d-k}(V)$ the set of subspaces intersecting $X$ transversally in $V$.

We need a simple fact from linear algebra, which we state in a rather general form that is useful for us in several places.

We need the following technical statement appearing in [Reference AleskerAle10, Proposition 5.1.3].

Proof. Let $(p_t,\xi _t)$ be a smooth path in $T^{*}M\setminus 0$. We lift it to a smooth path $(p_t,E_t,\xi _t,\eta _t)\in T^{*}(M\times \operatorname {Gr}_{d-k}(V))$ such that $p_t \in E_t$, and $(\xi _t,\eta _t)\in N^{*}_{p_t,E_t}Z_M$. Now for $v\in T_pM$, $B \in T_E \operatorname {Gr}_{d-k}(V)= \mathrm {Hom}(E,V/E)$, we have by Lemma 3.1 (applied with $l=1, L_0=\mathbb {R} p$) that $(v,B)\in T_{p,E}Z_M$ if and only if $v+E=B(p)$.

Hence,

\begin{align*} N^{*}_{p,E} Z_M=\{(\xi,\eta) \in T_p^{*}M \times T_E^{*}\operatorname{Gr}_{d-k}(V): \langle \xi,v\rangle+\langle \eta,B\rangle=0\text{ whenever } v+E=B(p)\}. \end{align*}

Fix a Euclidean structure on $V$, inducing Euclidean structures on the spaces $\mathrm {Hom}(E_t, V/E_t)$. Let us choose some $E_t$ such that $p_t\in E_t$, and $T_{p_t}M\cap E_t\subset \operatorname {Ker}(\xi _t)$, which evidently can be done. Consider the linear subspace

\[ W_t=\{ B \in T_{E_t}\operatorname{Gr}_{d-k}(V): B(p_t)\in (T_{p_t}M+E_t)/E_t\}, \]

and recall the natural isomorphism $q_t:(T_{p_t}M+E_t)/E_t \xrightarrow {\sim } T_{p_t}M/(T_{p_t}M\cap E_t)$. We now may define $\eta _t\in W_t^{*}$ by $\langle \eta _t,B\rangle =-\langle \xi _t,q_t(B(p_t))\rangle$ for each $B \in W_t$, as $T_{p_t}M\cap E_t \subset \ker \xi _t$. Extend $\eta _t$ by zero to $W_t^{\perp }$. It follows that $(\xi _t,\eta _t)\in N^{*}_{p_t,E_t}Z_M$, completing the proof.

It follows by [Reference AleskerAle10, Corollary 4.1.7] that the Radon transforms with respect to the Euler characteristic, $\mathcal {R}_M=(\tau _M)_*\pi _M^{*}:\mathcal {V}_c^{-\infty }(M)\to \mathcal {V}^{-\infty }(\operatorname {Gr}_{d-k}(V))$ and $\mathcal {R}_M^{T}=(\pi _M)_*\tau _M^{*}:\mathcal {V}^{\infty }(\operatorname {Gr}_{d-k}(V))\to \mathcal {V}^{\infty }(M)$, are well-defined and continuous.

Definition 3.4 For any $\phi \in \mathcal {V}_c^{-\infty }(M)$, let $\widehat \phi \in C^{-\infty }(\operatorname {Gr}_{d-k}(V))$ be the defining current of $\mathcal {R}_M\phi$ (on the base manifold). Equivalently, using [Reference AleskerAle07, Proposition 7.3.6] we have

\[ \widehat{\phi}=[\mathcal{R}_M\phi]\in\mathcal{W}_0^{-\infty}(\operatorname{Gr}_{d-k}(V))/\mathcal{W}_1^{-\infty}(\operatorname{Gr}_{d-k}(V))=C^{-\infty}(\operatorname{Gr}_{d-k}(V)). \]

Definition 3.6 The Crofton map $\operatorname {Cr}_M: \mathcal {M}^{\infty }(\operatorname {Gr}_{d-k}(V))\to \mathcal {W}_k^{\infty }(M)$ is the restriction of $\mathcal {R}_M^{T}$ to $\mathcal {M}^{\infty }(\operatorname {Gr}_{d-k}(V))$. More explicitly,

\[ \operatorname{Cr}_M(\mu)(X)=\int_{\operatorname{Gr}_{d-k}(V)}\widehat{\chi}_X(E)\,d\mu(E), \quad X \in \mathcal{P}(M). \]

We show in Proposition 3.12 that $\widehat {\chi }_X(E)=\chi (X \cap E)$.

3.2 Distributional Crofton measures

To allow distributional Crofton measures, it seems essential to require that all intersections $E\cap M$ are transversal, for $E\in \operatorname {Gr}_{d-k}(V)$. This is easily seen to be equivalent, for any $k>0$, to having $\mathbb {R} x\oplus T_xM=V$ for all $x\in M$. In particular, $\dim V=\dim M+1=m+1$. We deduce that $M$ is a hypersurface that is locally diffeomorphic to an open subset of $\mathbb {P}_+(V)$ through the radial projection. In other words, $M$ is locally a strictly star-shaped hypersurface around the origin.

Proof. (i) Let us first show $E\mapsto \psi (E\cap M)$ is smooth. By choosing an open cover of $M$ by star-shaped charts, and using the partition of unity property of smooth valuations [Reference AleskerAle07], we may assume $M$ projects diffeomorphically to an open subset of $\mathbb {P}_+(V)$, which we henceforth identify with $M$.

By Boman's theorem [Reference BomanBom67], it suffices to prove that $\psi (E_t \cap M)$ is a smooth function of $t\in (-\epsilon,\epsilon )$ for all smooth curves $E_\bullet :(-\epsilon,\epsilon )\to \operatorname {Gr}_{m+1-k}(V)$. It suffices, in fact, to show smoothness in some open interval around $0$ for any such given curve.

Let us lift $E_t$ to a smooth curve $g_t\in \mathrm {GL}(V)$ with $g_0=\operatorname {Id}$ and $E_t=g_tE_0$. Then $\psi (E_t\cap M)= g_t^{*}\psi (E_0 \cap M)$ for sufficiently small $t$ such that $g_t(\operatorname {Supp}(\psi ))\subset M$, establishing the first part.

(ii) Let us check $\psi (\bullet \cap M)=\widehat {\psi }$ in $C^{-\infty }(\operatorname {Gr}_{m+1-k}(V))$. Take $\mu \in \mathcal {M}^{\infty }(\operatorname {Gr}_{m+1-k}(V))$, and write

\[ \mu=\int_{\operatorname{Gr}_{m+1-k}(V)}\delta_E\,d\mu(E)= \int_{\operatorname{Gr}_{m+1-k}(V)}\chi_{\{E\}}\,d\mu(E)\in\mathcal{V}^{\infty}(\operatorname{Gr}_{m+1-k}(V)). \]

To see this, write $Z=Z_M$ and identify $W:=Z\times _M\mathbb {P}_M$ with its image in $\mathbb {P}_{Z}$ under $d\pi _M^{*}$. Explicitly, $W|_{(x,E)}=\mathbb {P}_+(\mathrm {Ker}(d_{(x,E)}\pi _M)^{\perp })$, so $W$ is the union of the conormal bundles to all fibers of $\pi _M$. It follows from [Reference AleskerAle10, Proposition 3.3.3] that $\operatorname {WF}(\pi _M^{*}\psi )\subset (\emptyset, N^{*}W)$. By Proposition 2.1, it suffices to check that two intersections in $\mathbb {P}_{Z}$ are transversal: $\pi ^{-1}(\tau _M^{-1}E)\pitchfork W$ and ${N^{*}(\tau _M^{-1}E)\pitchfork W}$.

Denote $z=(x,E)$, let $(z,\zeta )$ be an intersection point. The first intersection is easy to analyze: $T_{z,\zeta }\pi ^{-1}(\tau _M^{-1}E)$ contains all vertical directions of $\mathbb {P}_{Z}$, whereas $T_{z,\zeta }W$ contains all horizontal directions.

To analyze the second intersection, we lift all manifolds from $\mathbb {P}_Z$ to $T^{*}Z$, and retain all notation for the corresponding objects. As in the previous case, the image of $T_{z,\zeta }W$ under the natural projection $\pi :T_{z,\zeta } T^{*}Z\to T_zZ$ is all of $T_zZ$, and so it suffices to show $T_{z,\zeta }(T_z^{*}Z)\subset T_{z,\zeta }W+ T_{z,\zeta }N^{*}(\tau _M^{-1}E)$.

As $N_z^{*}(\pi _M^{-1}x) \subset W$, it suffices to show that

\[ T_{z,\zeta}(T_z^{*}Z)\subset T_{z,\zeta}N_z^{*}(\pi_M^{-1}x)+ T_{z,\zeta}N_z^{*}(\tau_M^{-1}E), \]

which is the same as

\[ T_z^{*}Z\subset N_z^{*}(\pi_M^{-1}x)+ N_z^{*}(\tau_M^{-1}E)=N_z^{*}(T_z\pi_M^{-1}x\cap T_z\tau_M^{-1}E). \]

The proof of the claim is completed by noting that the intersection $T_z\pi _M^{-1}x\cap T_z\tau _M^{-1}E$ is trivial.

Consider the set

\[ X:=\left\{(E,[\xi]): E \in \operatorname{Gr}_{m+1-k}(V), [\xi] \in \operatorname{WF}(\chi_{\tau_M^{-1}E})\right\} \subset \operatorname{Gr}_{m+1-k}(V) \times \mathbb{P}_{\mathbb{P}_Z}. \]

We claim that it is compact. If $X \subset \bigcup _{i \in I} U_i$ is an open cover, then for each $E \in \operatorname {Gr}_{m+1-k}(V)$ we find a finite subcover $X_E \subset \bigcup _{i \in I_E} U_i$ of the compact set

\[ X_E:=X\cap (\{E\} \times \mathbb{P}_{\mathbb{P}_Z})=\{E\} \times \operatorname{WF}(\chi_{\tau_M^{-1}E}). \]

The map $g \mapsto X_{gE}$ is $\operatorname {GL}(V)$-equivariant, hence there exists some open neighborhood $V_E \subset \operatorname {Gr}_{m+1-k}(V)$ of $E$ such that $X_{E'} \subset \bigcup _{i \in I_E} U_i$ for all $E' \subset V_E$. Now $\operatorname {Gr}_{m+1-k}(V)$ is compact, hence finitely many $V_{E_j}$ cover $\operatorname {Gr}_{m+1-k}(V)$. Then $X \subset \bigcup _j \bigcup _{i \in E_j} U_i$ is a finite subcover, proving the claim. The image of $X$ in $\mathbb {P}_{\mathbb {P}_Z}$ is then a compact set disjoint from $\operatorname {WF}(\pi _M^{*}\psi )$.

Thus, we can find a closed cone $\Gamma \subset T^{*}\mathbb {P}_{Z}\setminus \underline 0$ such that for all $E\in \operatorname {Gr}_{m+1-k}(V)$, $\chi _{\tau _M^{-1}E}\in \mathcal {V}^{-\infty }_{(\emptyset, \Gamma )}(Z)$, and $\pi _M^{*}\psi$ acts as a sequentially continuous functional on the latter space. Thus, we can write

\begin{align*} \langle \widehat{\psi},\mu\rangle&= \langle \pi_M^{*}\psi, \tau_M^{*}\int_{\operatorname{Gr}_{m+1-k}(V)} \chi_{\{E\}}\,d\mu(E)\rangle\\ & = \int_{\operatorname{Gr}_{m+1-k}(V)} \langle \pi_M^{*}\psi,\chi_{\tau_M^{-1}(E)}\rangle \,d\mu(E). \end{align*}

It remains to check that

(3)\begin{equation} \langle \pi_M^{*}\psi,\chi_{\tau_M^{-1}(E)}\rangle= \psi(E\cap M). \end{equation}

For a compact submanifold with boundary $A \subset M$ that is transversal to $E\cap M$, we have by [Reference Alesker and BernigAB12, Theorem 5],

\[ \langle \pi_M^{*}\chi_A,\chi_{\tau_M^{-1}(E)}\rangle=\chi(\pi_M^{-1}A\cap \tau_M^{-1}(E))=\chi(A\cap E)=\chi_A(E\cap M). \]

It follows by linearity that any smooth valuation of the form $\psi =\int _{\mathcal {A}} \chi _A\,d\nu (A)$, where $\mathcal {A}$ is a family of submanifolds $A$ as previously and $\nu$ a smooth measure, satisfies (3). This family $\mathbf {Cr}_E$ of valuations spans a dense subset in $\mathcal {V}^{\infty }(M)$. Indeed, we may approximate $\chi _A$ in $\mathcal {V}^{-\infty }(M)$ by a sequence in $\mathbf {Cr}_E$ for any $A$ transversal to $E\cap M$. Were $\mathbf {Cr}_E$ not dense, by Alesker–Poincaré duality one could find a non-zero smooth valuation $\phi$ annihilating $\mathbf {Cr}_E$ and, thus, also vanishing on all submanifolds with boundary that are transversal to $E\cap M$. By the genericity of transversality and continuity, $\phi$ would vanish on all submanifolds with boundary. However, this is impossible by [Reference Bernig and BröckerBB07]. It follows that equality in (3) holds for all $\psi$.

Proposition 3.9 The map $\operatorname {Cr}:\mathcal {M}^{\infty }(\operatorname {Gr}_{m+1-k}(V))\to \mathcal {W}_k^{\infty }(M)$ extends to a continuous map $\operatorname {Cr}: \mathcal {M}^{-\infty }(\operatorname {Gr}_{m+1-k}(V))\to \mathcal {W}_k^{-\infty }(M)$, by setting, for all $\psi \in \mathcal {V}^{\infty }_c(M)$,

\[ \left\langle \operatorname{Cr}(\mu), \psi\right\rangle := \int_{\operatorname{Gr}_{m+1-k}(V)} \psi(E\cap M)\,d\mu(E). \]

Proof. The right-hand side is well-defined for a generalized measure $\mu$, because the function $E \mapsto \psi (E \cap M)$ is smooth by Proposition 3.7. Take $\mu \in \mathcal {M}^{\infty }(\operatorname {Gr}_{m+1-k}(V))$, $\psi \in \mathcal {V}_c^{\infty }(M)$. To verify this new definition extends the smooth one, we ought to check that

\[ \langle (\pi_M)_*\tau_M^{*}\mu, \psi\rangle = \int_{\operatorname{Gr}_{m+1-k}(V)} \psi(E\cap M)\,d\mu(E), \]

which is the content of Corollary 3.8. Continuity is equally evident.

3.3 Functorial properties of Crofton measures

The following is a partial summary of the results of [Reference FaifmanFai17, Appendix B] (adapted from the affine to the linear Grassmannian), whereto we refer the reader for further details.

Let $j: U^{r}\hookrightarrow V^{d}$ be an inclusion of a linear subspace. There is then a well-defined operation of restriction

\[ j^{*}:\mathcal{M}^{\infty}(\operatorname{Gr}_k(V)) \to \mathcal{M}^{\infty}(\operatorname{Gr}_{k-(d-r)}(U)), \]

which is the push-forward under the (almost everywhere defined) map ${J_U: E\mapsto j^{-1}(E)=E\cap U}$.

Let $S_U\subset \operatorname {Gr}_{k}(V)$ be the collection of subspaces intersecting $U$ non-generically, and fix a closed cone $\Gamma \subset T^{*}\operatorname {Gr}_k(V)\setminus 0$ such that $\Gamma \cap N^{*}S_U=\emptyset$. Given $k\geq d-r$, let $\mathcal {M}^{-\infty }_\Gamma (\operatorname {Gr}_k(V))$ denote the set of generalized measures (distributions) $\mu$ whose wave front sets lie in $\Gamma$, equipped with the Hörmander topology.

The map $j^{*}$ extends as a sequentially continuous map

\[ j^{*}:\mathcal{M}_\Gamma^{-\infty}(\operatorname{Gr}_k(V)) \to \mathcal{M}^{-\infty}(\operatorname{Gr}_{k-(d-r)}(U)). \]

Similarly, if $\pi :V\to W$ is a quotient map, there is a natural push-forward operation

\[ \pi_*: \mathcal{M}^{\infty}(\operatorname{Gr}_k(V))\to \mathcal{M}^{\infty}(\operatorname{Gr}_{k}(W)), \]

which is the push-forward under the (almost everywhere defined) map $\Pi _W: E\mapsto \pi (E)$. It extends to distributions whose wave front sets are disjoint from the conormal cycle of the collection of subspaces intersecting $\operatorname {Ker} \pi$ non-generically.

The following proposition captures the intuitively obvious fact that the pullback of distributions/valuations under embeddings commutes with the Crofton map. We prove a weak version which suffices for our purposes.

Recall that $M$ is a locally star-shaped hypersurface around the origin.

Proof. Choose an approximate identity $\rho _i\in \mathcal {M}^{\infty }(\textrm {GL}(V))$ as $i\to \infty$, and set $\mu _i=\mu \ast \rho _{i}\in \mathcal {M}^{\infty }(\operatorname {Gr}_{d-k}(V))$. For all $A\in \mathcal {P}(Z)$ we have

\[ \operatorname{Cr}_M(\mu_i)(A)= \int_{\operatorname{Gr}_{d-k}(V)} \chi(A \cap E)\,d\mu_i(E)=\int_{\operatorname{Gr}_{r-k}(U)} \chi(A \cap E)d((J_U)_*\mu_i)(E), \]

and, therefore, $\operatorname {Cr}_M(\mu _i)|_Z=\operatorname {Cr}_Z(j^{*}\mu _i)$. The restriction of valuations to a submanifold is continuous in the Hörmander topology on the space of valuations with wave front set contained in $\textrm {WF}(\operatorname {Cr}_M(\mu ))$, see [Reference AleskerAle10, Claim 3.5.4]. Thus, the left-hand side weakly converges to $\operatorname {Cr}_M(\mu )|_Z$. The right-hand side weakly converges to $\operatorname {Cr}_Z(\mu )$.

3.4 Applying generalized Crofton formulas to subsets

Let $M^{m}\subset V=\mathbb {R}^{m+1}$ be a strictly star-shaped hypersurface around the origin. Given $A\in \mathcal {P}(M)$ and a Crofton distribution $\mu \in \mathcal {M}^{-\infty }(\operatorname {Gr}_{m+1-k}(V))$, we would like to evaluate $\operatorname {Cr}(\mu )$ on $A$ using an explicit Crofton integral, whenever $A\pitchfork \operatorname {Cr}(\mu )$.

The following proposition provides some a priori regularity for $\widehat \chi _A$.

Proof. Let us first check that $\chi (A \cap \bullet )\in L^{1} ( \operatorname {Gr}_{m+1-k}(V))$. Fix a Euclidean structure on $V$ and identify $M$ with the unit sphere. By [Reference Bernig, Fu and SolanesBFS14, Lemma A.2], for a fixed $E_0\in \operatorname {Gr}_{m+1-k}(V)$ we have $[g \mapsto \chi (A\cap gE_0)] \in L^{1}(\operatorname {SO}(V))$. Let $dg, dE$ be the Haar measures on $\operatorname {SO}(V)$ and $\operatorname {Gr}_{m+1-k}(V)$, respectively, and $p:\operatorname {SO}(V)\to \operatorname {Gr}_{m+1-k}(V)$ given by $g\mapsto gE_0$. Then $p_*(\chi (A\cap gE_0)dg)=\chi (A\cap E)\,dE$, and so $\chi (A\cap E)$ is integrable. It is evidently finite and locally constant on $W_A$

It remains to check that $\widehat {\chi }_A=\chi (A\cap \bullet )$. Take an approximate identity ${ \rho }_j\in \mathcal {M}^{\infty }(\operatorname {SO}(V))$, which for convenience we assume invariant under inversion.

Consider the convolution $\phi _j:=\chi _A\ast { \rho _j}\in \mathcal {V}^{-\infty }(M)$. As $\operatorname {SO}(V)$ is transitive on $M$ and $\mathbb {P}_M$, it follows that the defining currents of $\phi _j$ are smooth, and therefore $\phi _j\in \mathcal {V}^{\infty }(M)$.

By [Reference Bernig, Fu and SolanesBFS14, Theorem A.1], $\tilde \phi _j:=\int _{\operatorname {SO}(V)}\chi (gA\cap \bullet )\,d{\rho }_j(g)$ is a well-defined smooth valuation. Let us show that $\tilde \phi _j=\phi _j$. Take $\psi \in \mathcal {V}_c^{\infty }(M)$ and compute

\[ \langle\tilde\phi_j, \psi\rangle= \int_{\operatorname{SO}(V)}\psi(gA\cap M)\,d{ \rho}_j(g) =\int_{\operatorname{SO}(V)}\psi(gA)\,d{ \rho}_j(g) \]

by [Reference Bernig, Fu and SolanesBFS14], whereas

\[ \langle \phi_j, \psi\rangle=\langle \chi_A, \psi\ast { \rho}_j\rangle =(\psi\ast\nu_j)(A)=\int_{\operatorname{SO}(V)}\psi(gA)\,d{ \rho}_j(g). \]

Equality now follows by Alesker–Poincaré duality. We thus have the following equalities of functions on $\operatorname {Gr}_{m+1-k}(V)$:

\[ \phi_j( { \bullet} \cap M)=\tilde\phi_j({ \bullet}\cap M)=\int_{\operatorname{SO}(V)}\chi(gA\cap { \bullet})\,d{ \rho}_j(g)=\chi(A\cap \bullet)\ast { \rho}_j, \]

where the right-hand side is the convolution of $\chi (A\cap \bullet )\in L^{1}(\operatorname {Gr}_{m+1-k}(V))$ with ${ \rho }_j$. It follows that $\phi _j(E\cap M)\to \chi (A\cap E)$ in $L^{1}(\operatorname {Gr}_{m+1-k}(V))$.

Fix $\mu \in \mathcal {M}^{\infty }(\operatorname {Gr}_{m+1-k}(V))$. By Proposition 3.7 and $\operatorname {GL}(V)$-equivariance,

\begin{align*} \langle \widehat{\chi}_A,\mu \rangle &= \lim_{j\to\infty} \langle \widehat{\chi}_A \ast { \rho}_j, \mu\rangle = \lim_{j\to\infty} \langle \widehat {\chi_A\ast { \rho}_j}, \mu\rangle\\ &=\lim_{j\to\infty} \int_{\operatorname{Gr}_{m+1-k}(V)} \phi_j(E\cap M) \,d\mu(E) =\int_{\operatorname{Gr}_{m+1-k}(V)} \chi(A\cap E) \,d\mu(E), \end{align*}

and so $\widehat {\chi }_A=\chi (A\cap \bullet )$.

As $\widehat {\chi }_A$ is real-valued, $\operatorname {Cr}\operatorname {WF}^{k}(A)$ must be symmetric under the fiberwise antipodal map.

Proposition 3.14 Assume $\operatorname {Cr}\operatorname {WF}^{k}(A)\cap \operatorname {WF}(\mu )=\emptyset$. Then

(4)\begin{equation} \operatorname{Cr}(\mu)(A)=\int_{\operatorname{Gr}_{m+1-k}(V)}\chi(A\cap E)\,d\mu(E). \end{equation}

Proof. We identify $M$ with $\mathbb {P}_+(V)$. Then $\mathcal {V}^{-\infty }(M)\to C^{-\infty }(\operatorname {Gr}_{m+1-k}(V))$, $\phi \mapsto \widehat \phi$ is $\operatorname {GL}(V)$-equivariant. Consider the sequence of smooth valuations $\psi _j$ given by $\psi _j=\int _{\operatorname {GL}(V)}g^{*}\chi _A\cdot d \rho _j(g)$, where $\rho _j$ is a compactly supported approximate identity on $\operatorname {GL}(V)$. Clearly $\psi _j\to \chi _A$ in the Hörmander topology of $\mathcal {V}^{-\infty }_{\operatorname {WF}(\chi _A)}(M)$. By $\operatorname {GL}(V)$-equivariance we have that

\begin{align*} \widehat{\psi}_j= \int_{\operatorname{GL}(V)}g^{*}\widehat{\chi}_A \cdot d \rho_j(g) \to\widehat\chi_A \end{align*}

in $C^{-\infty }_{\operatorname {WF}(\widehat \chi _A)}(\operatorname {Gr}_{m+1-k}(V))$.

It holds by Propositions 3.9 and 3.7 that

\[ \langle \operatorname{Cr}(\mu), \psi_j\rangle = \int_{\operatorname{Gr}_{m+1-k}(V)} \psi_j(E\cap M)\,d\mu(E)=\langle \mu,\widehat{\psi}_j\rangle. \]

As $j \to \infty$, the left-hand side converges to $\langle \operatorname {Cr}(\mu ),\chi _A\rangle =\operatorname {Cr}(\mu )(A)$, as $A \pitchfork \operatorname {Cr}(\mu )$. The right-hand side converges to $\langle \mu,\widehat {\chi }_A\rangle$ (because $\operatorname {WF}(\mu ) \cap \operatorname {WF}(\widehat {\chi }_A)=\emptyset$), which is the same as $\int _{\operatorname {Gr}_{m+1-k}(V)}\chi (A\cap E)\,d\mu (E)$ by Proposition 3.12.

Determining $\operatorname {Cr}\operatorname {WF}^{k}(A)$ precisely appears to be difficult in general. Let us focus on a subset $A\in \mathcal {P}(M)$ which is either a compact domain with smooth boundary, or a compact hypersurface without boundary.

For the following, we write $H=H(A)$ for $\partial A$ if $A$ is of full dimension, and for $A$ when it is a hypersurface. Write $\widehat E:=E\cap M$, and note that $E$ intersects $H$ transversally in $V$ if and only if $\widehat E$ intersects $H$ transversally in $M$. We use the notation

\begin{align*} \widetilde B_H & :=\{(x,E)\in Z_H: T_x\widehat E\subset T_xH\}, \\ B_H & : =\tau_H(\widetilde B_H)\subset\operatorname{Gr}_{m+1-k}(V). \end{align*}

It is not hard to see that $\widetilde B_H$ is an embedded submanifold of $Z_H$ of dimension

(5)\begin{align} \dim \widetilde B_H&= \dim H+(m-k)(\dim H-(m-k))\nonumber\\ &=k(m+1-k)-1=\dim \operatorname{Gr}_{m+1-k}(V)-1. \end{align}

If $(x,E)\in \widetilde B_H$, we say that $x\in H$ is a tangent point for $E$. Observe also that $W_A= B_H^{c}$.

Write $\tilde \tau _H$ for the restriction $\tau _H|_{\widetilde B_H}:\widetilde B_H\to \operatorname {Gr}_{m+1-k}(V)$. We sometimes write $B^{m+1-k}_H$, etc., to specify the dimension.

Note that if $E\notin B_H$, then it is automatically regular.

For a subset $A\subset V$ we denote by $\mathbb {P}(A)$ its image in the projective space $\mathbb {P}(V)$. The regularity of the tangent is equivalent to the non-vanishing of the Gauss curvature of the corresponding section, as follows.

We now describe the Crofton wave front near regular tangents. For an immersed manifold $i:X\looparrowright Y$ and $y\in i(X)$, we denote

\[ N_y^{*}i(X)=\bigcup_{x\in i^{-1}y}(d_xi(T_xX))^{\perp}\subset T_y^{*}Y,\quad N^{*}i(X)=\bigcup_{y\in i(X)}N^{*}_yi(X). \]

That $\operatorname {Cr}\operatorname {WF}^{k}_{E_0}(A)$ is contained in the sum of the conormal spaces of the embedded parts of $B_H$ follows from the fact that $\widehat \chi _A$ is locally constant on the complement of $B_H$. However, to show that it is actually contained in the union of those conormal spaces, in the following proof we need a more precise description of $\widehat \chi _A$.

Proof. In the following, by a ball (centered at a point) we mean a compact contractible neighborhood (of the point) with smooth boundary. As $\widetilde B_H$ is a submanifold of $Z_H$, by assumption $B_H\subset \operatorname {Gr}_{m+1-k}(V)$ is an immersed submanifold in a neighborhood around $E_0$, which is a hypersurface by (5).

The preimage $\tilde \tau _H^{-1}(E_0)$ must be finite, or otherwise we could find a sequence of distinct points $(q_j, E_0)\in \widetilde B_H$, which then has a limit point $(q_0, E_0)$, and $\tilde \tau _H$ would fail to be injective in a neighborhood of $(q_0, E_0)$, contradicting the assumed immersivity of $\tilde \tau _H$ there. We use the notation $\tilde \tau _H^{-1}(E_0)=\{(q_j, E_0), 1\leq j\leq N\}$.

We can now find a ball $W \subset \operatorname {Gr}_{m+1-k}(V)$ centered at $E_0$, such that $B_H\cap W$ is the finite union of embedded hypersurfaces $F_j$, each diffeomorphic to a Euclidean ball, with $E_0\in F_j$ and $\partial F_j\subset \partial W$ for all $j$. Note that we have no control on how these hypersurfaces intersect each other. Denote by $C_j^{\pm }$ the connected components of $W\setminus F_j$. The indices are matched by requiring that a neighborhood of $(q_j, E_0)\in \widetilde B_H$ is mapped to $F_j$ by $\tilde \tau _H$.

Fix small balls $K_j\subset M$ around $q_j$ such that

\[ \tilde\tau_H:\pi_H^{-1}(K_j)\cap \tilde\tau_H^{-1}(W) \to F_j \]

is an embedding, $\partial K_j\pitchfork H$ and $\partial K_j\pitchfork \widehat E_0$ in $M$. As $Z_0:=\{q_j:1\leq j\leq N\}$ is the subset of all points in $H$ where $\widehat E_0$ fails to intersect $H$ transversally, it holds that $\widehat E_0\pitchfork (H\setminus Z_0)$, and so $\widehat E_0\cap (H\setminus Z_0)$ is a locally closed submanifold in $H$. We assume the $K_j$ small enough so that they are pairwise disjoint, and in particular $\partial K_j\cap Z_0=\emptyset$. We may, moreover, assume that $H\cap K_j$ is diffeomorphic to a Euclidean ball. By the transversality theorem, we may perturb $K_j$ if necessary to have $(\widehat E_0\cap H) \pitchfork (\partial K_j \cap H)$ in $H$.

Denote by $\tfrac {1}{2} K_j$ a smaller ball centered at $q_j$. Taking $W$ sufficiently small, we may assume that

(6)\begin{equation} \widehat E \pitchfork \biggl(H\setminus \bigcup_j \frac 12 K_j\biggr),\quad\forall E\in W. \end{equation}

This follows by the stability of transversal intersections, because $\widehat E$ is a smooth perturbation of $\widehat E_0$, which intersects $H$ transversally in an open neighborhood of $H\setminus \bigcup _j \textrm {int}(\frac 12 K_j)$. Similarly, we have

(7)\begin{align} \widehat E \pitchfork \partial K_j\textrm{ in } M,\quad \forall E\in W, 1\leq j\leq N \end{align}

and

(8)\begin{equation} (\widehat E\cap H) \pitchfork (\partial K_j\cap H) \textrm{ in } H,\quad \forall E\in W, 1\leq j\leq N. \end{equation}

For $\epsilon \in \{\pm \}^{N}$, denote $C_\epsilon =\cap _{j=1}^{N} C_j^{\epsilon _j}$. Recall that $\widehat {\chi }_A$ is locally constant on $W_A=B_H^{c}$, and so is constant on any connected component of a non-empty set $C_\epsilon$. Let us show there are integers $e_j=e_j(E_0)$ such that for any $\epsilon,\epsilon ' \in \{\pm \}^{N}$ and any $E\in C_{\epsilon }, E'\in C_{\epsilon '}$ one has

(9)\begin{equation} \widehat{\chi}_A(E')-\widehat\chi_A(E)=\sum_{j:\epsilon_j<\epsilon_j'}e_j-\sum_{j:\epsilon_j'<\epsilon_j}e_j. \end{equation}

For $E\in W$, denote $\Sigma _j(E):=\widehat E\cap \partial K_j\cap H$. As it is the transversal intersection of $\widehat E\cap H$ and $\partial K_j \cap H$ in $H$, it is a closed manifold of dimension $(m-k-2)$, and $\chi (\Sigma _j(E))$ is independent of $E\in W$.

Let us distinguish the two cases under consideration. Assume first $A=H$ is a hyper- surface. As $K_i \cap K_j = \emptyset$, we have

\[ \mathbb{1}_M=\sum_{j=1}^{N} (\mathbb{1}_{K_j}-\mathbb{1}_{\partial K_j})+\mathbb{1}_{\overline {K^{c}}}, \]

with $K:=\bigcup _{j=1}^{N} K_j$. Hence, for $E\in W\setminus B_H$ we have

\[ \chi(E\cap A) =\sum_{j=1}^{N} \chi(E\cap K_j\cap A) -\sum_{j=1}^{N}\chi(\Sigma_j(E))+\chi(E\cap \overline{K^{c}} \cap A). \]

The last summand is constant on $W$ by properties (6) and (8). Consequently, for $E\in C_\epsilon$, $E'\in C_{\epsilon '}$ we have

\begin{align*} \widehat\chi_A(E')-\widehat{\chi}_A(E)=\sum_{j=1}^{N} \big(\chi(E'\cap A\cap K_j)- \chi(E\cap A\cap K_j)\big). \end{align*}

The function $\chi (\bullet \cap A\cap K_j)$ is locally constant on $W\setminus F_j$, and it remains to define

(10)\begin{equation} e_j:=\chi(\bullet\cap A\cap K_j)|_{C_j^{+}}- \chi(\bullet\cap A\cap K_j)|_{C_j^{-}}. \end{equation}

The case of full-dimensional $A$ is only slightly more involved. If $(m-k)$ is odd, we have $\widehat {\chi _A}=\tfrac 12\widehat {\chi _{\partial A}}$, reducing to the previous case. Thus, assume $(m-k)$ is even.

Write, as before, whenever $E\in W\setminus B_H$,

\[ \chi(E\cap A)=\sum_{j=1}^{N} \chi(E\cap K_j\cap A) -\sum_{j=1}^{N}\chi(E\cap \partial K_j\cap A)+\chi(E\cap \overline{K^{c}}\cap A). \]

Note that for $E\in W\setminus B_H$, all intersections are manifolds with corners. We have

\[ \chi(E\cap \partial K_j\cap A)=\tfrac 12\chi(E\cap \partial K_j\cap \partial A)=\tfrac 12\chi(\Sigma_j(E)) \]

thus it is constant in $W$.

Set

\[ S_j:=\widehat E\cap \partial K_j, \quad S:=\widehat E\cap \partial A. \]

Here $S_j$ is a transversal intersection in $M$ for all $E\in W$ and, hence, a smooth hyper- surface, whereas $S$ is given by a transversal intersection in $M$ and, hence, smooth for $E\in W\setminus B_H$. Moreover, $S$ is a smooth hypersurface outside of $\tfrac 12 K_j$ for all $E\in W$.

We claim that the intersection $S_j \cap S=\Sigma _j(E)$ is transversal in $\widehat E$ for all $E\in W$. If the intersection is not transversal at $x$, then $T_x( \widehat E\cap \partial K_j)=T_x( \widehat E\cap \partial A)$. However, by assumption $\widehat E \cap \partial A$ and $\partial K_j \cap \partial A$ intersect transversally in $\partial A$, in particular

\[ T_x(\widehat E \cap \partial A)+T_x(\partial K_j \cap \partial A)=T_x\partial A. \]

In conjunction with the previous equality, we get $T_x\partial A \subset T_x\partial K_j$, which is false.

Let $X, Y\subset P$ be smooth domains in a manifold $P$, and assume $\partial X\pitchfork \partial Y$ and $X$ is compact. Let $Z\subset X\cap Y$ be the closure of a connected component of $X\cap Y$. Then $\chi (Z)$ is constant as $X, Y$ are perturbed while maintaining transversality.

Taking $P=\widehat E$, $X=E\cap A$ with $\partial X=S$ (which is a manifold for $E \in W \setminus B_H$), and $Y=E \cap K_j$ with $\partial Y=S_j$ we obtain that $\chi (E\cap A\cap K_j)$ is locally constant in $W\setminus B_H$. Taking $X=E\cap A$ with $\partial X=S$ (which is a manifold outside $\bigcup _j \tfrac 12 K_j$ for all $E \in W$), $Y=E\cap \overline {K^{c}}$ with $\partial Y=\bigcup _j S_j$, it follows that $\chi (E\cap A\cap \overline {K^{c}})$ is constant in $W$. Thus, we may define $e_j$ as in the previous case by (10).

It follows from (9) that for $E\in W$, $\widehat {\chi }_A(E)$ is a linear combination of the indicator functions of the connected components of the complements of the hypersurfaces $F_j$ in $W$. Therefore, $\operatorname {WF}_{E_0}(\widehat \chi _A)\subset \bigcup _j N^{*}F_j$, concluding the proof.

5.1 A holomorphic family of Crofton measures

For the following, let $\operatorname {Sym}_r^{+}(\mathbb {R})\subset \operatorname {Sym}_r(\mathbb {R})$ be the cone of positive-definite matrices, and $\mathfrak h_r=\operatorname {Sym}_r(\mathbb {R})\oplus \mathbf {i}\operatorname {Sym}_r^{+}(\mathbb {R})\subset \operatorname {Sym}_r(\mathbb {C})$ the Siegel upper half space. The following is well-known.

Recall for the following that given a non-degenerate quadratic form $Q$ on $V$, a compatible Euclidean form is any positive-definite form $P$ such that $V$ admits a decomposition $V=V_+\oplus V_-$ which is both $P$- and $Q$-orthogonal, and $Q|_{V_\pm }=\pm P|_{V_{\pm }}$.

From here on, let $V=\mathbb {R}^{p} \oplus \mathbb {R}^{q}=\mathbb {R}^{n+1}$ with the standard quadratic form $Q$ of signature $(p,q)$ and the corresponding compatible Euclidean form $P_0$. Define a family of complex-valued quadratic forms $Q_\zeta$ on $V$ with $\zeta \in \mathbb {C}$, by

\[ Q_\zeta:=Q+2\zeta P_0. \]

We then have

\[ Q_\zeta(x,y):= \begin{cases} (2\zeta+1) P_0(x,y) & x,y \in \mathbb{R}^{p},\\ (2\zeta-1) P_0(x,y) & x,y \in \mathbb{R}^{q},\\ 0 & x \in \mathbb{R}^{p}, y \in \mathbb{R}^{q}. \end{cases} \]

Observe that $Q_\zeta$ is real and positive-definite for $\zeta >\tfrac 12$, and $Q_0=Q$. Furthermore, by Lemma 5.1, $\det Q_\zeta \neq 0$ for $\zeta \in U_{\mathbb {C}}$, as either $Q_\zeta$ or $\mathbf {i} Q_\zeta$ lies in $\mathfrak h_{n+1}$. Note that a complex-valued non-degenerate quadratic form $Q$ on a real vector space $E$ defines an element $\operatorname {vol}_Q^{2}\in \operatorname {Dens}_\mathbb {C}(E)^{2}$, and given a branch of square root we also get a complex-valued density $\operatorname {vol}_Q\in \operatorname {Dens}_\mathbb {C}(E)$.

By a ($P$-)frame on an open subset $U\subset \operatorname {Gr}_{n+1-k}(V)$ we understand a smooth section of the Stiefel manifold of $P$-orthonormal $(n+1-k)$-frames in $V$ defined over $U$. For a subspace $E\in \operatorname {Gr}_{n+1-k}(V)$ and $\zeta \in U_{\mathbb {C}}$, choose a frame $u_i(E)$ on $U$ and define $X^{P}_\zeta :U\to \operatorname {Sym}_{n+1-k}(\mathbb {C})$ to be the corresponding Gram matrix of $Q_\zeta$, namely $X^{P}_\zeta (E)=(Q_\zeta (u_i(E), u_j(E)))_{i,j=1}^{n+1-k}$.

Note that if $\widetilde X_{\zeta }$ is the corresponding matrix for a different frame $\widetilde u_i(E)$ on $U$, then

(17)\begin{equation} \widetilde X_\zeta(E)=B(E)^{T} X_\zeta(E) B(E) \end{equation}

for some smooth map $B:U\to \operatorname {O}(n+1-k)$.

Observe that by (17), $\det (X^{P}_\zeta )$ is independent of the choice of $P$-orthonormal bases of $E$. Moreover, either the real or imaginary part of $Q_\zeta |_E$ is positive-definite, and consequently by Lemma 5.1, $\det X^{P}_\zeta (E)\neq 0$.

The function $\det (X^{P}_\zeta )^{\lambda }\in C^{\infty }(\operatorname {Gr}_{n+1-k}(V),\mathbb {C})$ is thus well-defined for all $P$ and $\lambda \in \mathbb {C}$, and analytic in $\zeta \in U_{\mathbb {C}}$, once the normalization $\det (X^{P}_1)^{\lambda }>0$ is fixed, as $U_{\mathbb {C}}$ is simply connected.

Define the smooth measure $\widetilde m^{\zeta, P}_k$ on the Grassmannian $\operatorname {Gr}_{n+1-k}(V)$ by

\[ d\widetilde m^{\zeta, P}_k:=\det(X^{P}_\zeta)^{-({n+1})/{2}}(E)\,d\sigma_P(E), \]

where $d\sigma _P(E)$ is the $\operatorname {O}(P)$-invariant probability measure on the Grassmannian.

Proposition 5.2 The complex-valued smooth measure

\[ m^{\zeta}_k:=(2\zeta+1)^{{p(n+1-k)}/{2}}(2\zeta-1)^{{q(n+1-k)}/{2}}\widetilde m_k^{\zeta,P_0}, \quad \zeta\in U_{\mathbb{C}} \]

depends analytically on $\zeta$ and is normalized, i.e.

\[ \int_{\operatorname{Gr}_{n+1-k}(V)} \,dm^{\zeta}_k=1. \]

Proof. The first statement is clear. For the second, we first see how $\widetilde m^{\zeta, P}_k$ depends on $P$. Let $P_1,P_2$ be two Euclidean structures on $V$. From the natural identification $T_E\operatorname {Gr}_{n+1-k}(V)=E^{*}\otimes V/E$ we obtain that

\[ \operatorname{Dens}(T_E\operatorname{Gr}_{n+1-k}(V))=\operatorname{Dens}^{*}(E)^{n+1}\otimes \operatorname{Dens}(V)^{n+1-k}. \]

Spelling this out gives

\[ \frac{d\sigma_{P_1}(E)}{d\sigma_{P_2}(E)}=\biggl(\frac{\operatorname{vol}_{P_1|_E}}{\operatorname{vol}_{P_2|_E}}\biggr)^{-(n+1)} \biggl(\frac{\operatorname{vol}_{P_1}}{\operatorname{vol}_{P_2}}\biggr)^{n+1-k}, \]

As

\[ \det X_\zeta^{P_i}(E)=\frac{\operatorname{vol}^{2}_{Q_\zeta|_E}}{\operatorname{vol}^{2}_{P_i|_E}}, \]

we find that

\[ \widetilde m^{\zeta, P_1}_k=\biggl(\frac{\operatorname{vol}_{P_1}}{\operatorname{vol}_{P_2}}\biggr)^{n+1-k}\widetilde m^{\zeta, P_2}_k. \]

For $\zeta >\tfrac {1}{2}$, $Q_\zeta$ is a Euclidean structure. Then

\begin{align*} 1 & =\int \widetilde m^{\zeta, Q_\zeta}_k= \biggl(\frac{\operatorname{vol}_{Q_\zeta}}{\operatorname{vol}_{P_0}}\biggr)^{n+1-k} \int \widetilde m^{\zeta, P_0}_k\\ & = \sqrt{2\zeta+1}^{p(n+1-k)} \sqrt{2\zeta-1}^{q(n+1-k)} \int \widetilde m^{\zeta, P_0}_k=\int m^{\zeta}_k. \end{align*}

By uniqueness of the analytic continuation, this formula also holds for general $\zeta \in U_{\mathbb {C}}$.

5.2 Homogeneous distributions on the space of symmetric matrices

Lemma 5.3 The meromorphic family of generalized functions

\[ f_\lambda(X):=\sum_{h=0}^{r} e^{\mathbf{i} \pi h\lambda}|\det X|^{\lambda}_{r-h} \in C^{-\infty}(\operatorname{Sym}_r(\mathbb{R})) \]

is analytic in $\lambda \in \mathbb {C}$ and satisfies

(18)\begin{equation} f_\lambda(-X)=e^{\mathbf{i} \pi r \lambda} \overline{f_\lambda(X)}. \end{equation}

Proof. We recall some results from [Reference MuroMur99]. Consider a linear combination

\[ g_\lambda(X):=\sum_{h=0}^{r} a_h |\det X|^{\lambda}_{r-h} \]

with constant coefficients $a_h \in \mathbb {C}$ and set $\vec a:=(a_0,\ldots,a_r) \in \mathbb {C}^{r+1}$. Then $g_\lambda \in C^{-\infty }(\operatorname {Sym}_r(\mathbb {R}))$ is meromorphic with possible poles in the set $\{-m, -({2m+1})/{2}: m\geq 1\}$.

The order of the pole at $s$ in this set can be obtained as follows. Set $\epsilon =-1$ if $s$ is an even integer and $\epsilon =1$ otherwise. Define inductively linear maps $d^{(m)}=(d^{(m)}_0,\ldots,d^{(m)}_{r+1-m}):\mathbb {C}^{r+1} \to \mathbb {C}^{r+1-m}$ by setting

\begin{align*} d^{(0)}_h(\vec a) & := a_h,\\ d^{(1)}_h(\vec a) & :=a_h+\epsilon a_{h+1} ,\\ d_h^{(2l+1)}(\vec a) & := d^{(2l-1)}_h-d^{(2l-1)}_{h+2}, \quad l=1,2,\ldots,\\ d_h^{(2l)}(\vec a) & := d^{(2l-2)}_h+d^{(2l-2)}_{h+2}, \quad l=1,2,\ldots. \end{align*}

Then $g_\lambda$ has a pole of order $p$ at $s=-({2m+1})/{2}$ if and only if $d^{2p}(\vec a) \neq 0, d^{2p+2}(\vec a)=0$. Similarly, $g_\lambda$ has a pole of order $p$ at $s=-m$ if and only if $d^{2p-1}(\vec a) \neq 0, d^{2p+1}(\vec a)=0$. Here we use the convention that $d^{(m)}=0$ if $m >r+1$ and that a pole of order $0$ is a point of analyticity.

In our situation, the coefficients $a_h=a_h(\lambda )=e^{\mathbf {i} \pi h \lambda }$ depend on $\lambda$ and we cannot apply Muro's result directly. However, writing

\[ f_\lambda(X)=\sum_{h=0}^{r} a_h(\lambda) |\det X|^{\lambda}_{r-h}=\sum_{j=0}^{\infty} \frac{(\lambda-s)^{j}}{j!} \sum_{h=0}^{r} a_h^{(j)}(s) |\det X|^{\lambda}_{r-h}, \]

we see that it is enough to prove that the order of the pole of $\sum _{h=0}^{r} a_h^{(j)}(s) |\det X|^{\lambda }_{r-h}$ at $\lambda =s$ is at most $j$ for all $j$.

By induction, we find that for all $l=0,1,\ldots$

\begin{align*} d^{2l}_h(\vec a(\lambda)) & = e^{\mathbf{i} \pi h \lambda} (1+e^{2\pi \mathbf{i} \lambda})^{l}, \\ d^{2l+1}_h(\vec a(\lambda)) & = e^{\mathbf{i} \pi h \lambda} (1+\epsilon e^{\mathbf{i} \pi \lambda}) (1-e^{2\pi \mathbf{i} \lambda})^{l}, \end{align*}

and, hence,

\begin{align*} d^{2j+2}_h\biggl(\vec a^{(j)}\biggl(-\frac{2m+1}{2}\biggr)\biggr) & = \frac{d^{j}}{d\lambda^{j}}\biggr|_{\lambda=-({2m+1})/{2}} e^{\mathbf{i} \pi h \lambda} (1+e^{2\pi \mathbf{i} \lambda})^{j+1}=0, \\ d^{2j+1}_h\big(\vec a^{(j)}(-m)\big) & = \frac{d^{j}}{d\lambda^{j}}\biggr|_{\lambda=-m} e^{\mathbf{i} \pi h \lambda} (1+\epsilon e^{\mathbf{i} \pi \lambda}) (1-e^{2\pi \mathbf{i} \lambda})^{j}=0, \end{align*}

which finishes the proof.