Geodesic
On infinitesimal $\boldsymbol { \tau }$-isospectrality of locally symmetric spaces
Canadian mathematical bulletin, Tome 68 (2025) no. 1, pp. 246-261

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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.
DOI : 10.4153/S0008439524000882
Mots-clés : Representation equivalence, isospectrality, Selberg trace formula, non-compact symmetric space 
Bhagwat, Chandrasheel; Mondal, Kaustabh; Sachdeva, Gunja. On infinitesimal $\boldsymbol { \tau }$-isospectrality of locally symmetric spaces. Canadian mathematical bulletin, Tome 68 (2025) no. 1, pp. 246-261. doi: 10.4153/S0008439524000882
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     title = {On infinitesimal $\boldsymbol { \tau }$-isospectrality of locally symmetric spaces},
     journal = {Canadian mathematical bulletin},
     pages = {246--261},
     year = {2025},
     volume = {68},
     number = {1},
     doi = {10.4153/S0008439524000882},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439524000882/}
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