Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Selberg Trace Formula IX: Contribution from the Conjugacy Classes (The Regular Case)
103
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper initiates a study into the contribution to the trace provided by the conjugacy classes.
rate research
Read More
Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test functions. In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C*-algebras of G and H. As an application, we exploit the role of group C*-algebras as recipients of higher indices of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours.
We study elements of the spectral theory of compact hyperbolic orbifolds $Gamma backslash mathbb{H}^{n}$. We establish a version of the Selberg trace formula for non-unitary representations of $Gamma$ and prove that the associated Selberg zeta function admits a meromorphic continuation to $mathbb{C}$.
We explicate the combinatorial/geometric ingredients of Arthurs proof of the convergence and polynomiality, in a truncation parameter, of his non-invariant trace formula. Starting with a fan in a real, finite dimensional, vector space and a collection of functions, one for each cone in the fan, we introduce a combinatorial truncated function with respect to a polytope normal to the fan and prove the analogues of Arthurs results on the convergence and polynomiality of the integral of this truncated function over the vector space. The convergence statements clarify the important role of certain combinatorial subsets that appear in Arthurs work and provide a crucial partition that amounts to a so-called nearest face partition. The polynomiality statements can be thought of as far reaching extensions of the Ehrhart polynomial. Our proof of polynomiality relies on the Lawrence-Varchenko conical decomposition and readily implies an extension of the well-known combinatorial lemma of Langlands. The Khovanskii-Pukhlikov virtual polytopes are an important ingredient here. Finally, we give some geometric interpretations of our combinatorial truncation on toric varieties as a measure and a Lefschetz number.
We define a map from the set of conjugacy classes of a Weyl group W to the representation ring of W tensored with the ring of polynomials in one variable.
A {it $k$-involution} is an involution with a fixed point set of codimension $k$. The conjugacy class of such an involution, denoted $S_k$, generates $text{Mob}(n)$-the the group of isometries of hyperbolic $n$-space-if $k$ is odd, and its orientation preserving subgroup if $k$ is even. In this paper, we supply effective lower and upper bounds for the $S_k$ word length of $text{Mob}(n)$ if $k$ is odd, and the $S_k$ word length of $text{Mob}^+(n)$, if $k$ is even. As a consequence, for a fixed codimension $k$ the length of $text{Mob}^{+}(n)$ with respect to $S_k$, $k$ even, grows linearly with $n$ with the same statement holding in the odd case. Moreover, the percentage of involution conjugacy classes for which $text{Mob}^{+}(n)$ has length two approaches zero, as $n$ approaches infinity.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
