Shelstad's character identity from the point of view of index theory

TL;DR

This paper uses index theory of elliptic operators within $K$-theory to geometrically prove Shelstad's character identity for discrete series, linking index theory to the Langlands classification of representations.

Contribution

It provides a geometric proof of Shelstad's character identity for discrete series using index theory, suggesting a new approach in the Langlands program.

Findings

Index theory can be used to prove character identities.

The geometric proof supports the role of index theory in representation classification.

Evidence of index theory's relevance in the Langlands program.

Abstract

Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding $L$-packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of $K$-theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program.