1 results
On infinitesimal
$\boldsymbol { \tau }$-isospectrality of locally symmetric spaces
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 68 / Issue 1 / March 2025
- Published online by Cambridge University Press:
- 09 January 2025, pp. 246-261
- Print publication:
- March 2025
-
- Article
- Export citation
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Let
$(\tau , V_{\tau })$ be a finite dimensional representation of a maximal compact subgroup K of a connected non-compact semisimple Lie group G, and let 
$\Gamma $ be a uniform torsion-free lattice in G. We obtain an infinitesimal version of the celebrated Matsushima–Murakami formula, which relates the dimension of the space of automorphic forms associated to 
$\tau $ and multiplicities of irreducible 
$\tau ^\vee $-spherical spectra in 
$L^2(\Gamma \backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal 
$\tau $-isospectrality. Along with this, we prove that the almost equality of 
$\tau $-spherical spectra of two lattices assures the equality of their 
$\tau $-spherical spectra.
