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Certain finiteness results for local Kac-Moody groups Ali, Abid


About half a century ago, Simon Gindikin and Fredrick Karpelevich evaluated the well known Harish Chandraâ à ôs \textbf{c}-function for semisimple Lie groups. This solution became known as the Gindikin-Karpelvich formula. While studying the constant term of Eisenstein series on adelic groups, Langland in \emph{Euler Products}, formulated the $p$-adic analogue of \textbf{c}-function and solved this integral. Macdonald independently obtained this formula for $p$-adic Chevalley groups in his lectures notes \emph{Spherical Functions on a Group of p-adic Type}. In Kac-Moody settings, which are infinite dimensional in general, the first challenge is to show that the algebraic analogue of the \textbf{c}-function is well defined. This can be done by proving certain finiteness results. For affine Kac-Moody groups, Braverman, Garland, Kazhdan, and Patnaik (BGKP) did this in 2014. Recently, Auguste H´ebert generalized these results by using the combinatorial objects called \emph{hovels} associated with Kac-Moody groups. We are trying to obtain these finiteness results using the algebraic methods motivated by the work of BGKP. In my talk, I will describe these results and share our progress on it. This is a joint project with Manish Patnaik.

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