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In this paper, we study the relation between the cocenter \overline {{ilde {\mathcal H}}} and the representations of an affine pro-\overline {{ilde {\mathcal H}}} Hecke algebra \overline {{ilde {\mathcal H}}} . As a consequence, we obtain a new criterion on supersingular representations: a (virtual) representation of \overline {{ilde {\mathcal H}}} is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter \overline {{ilde {\mathcal H}}} .

Zelevinsky’s classification theory of discrete series of p-adic general linear groups has been well known. Mœglin and Tadić gave the same kind of theory for p-adic classical groups, which is more complicated due to the occurrence of nontrivial structure of L-packets. Nonetheless, their work is independent of the endoscopic classification theory of Arthur (also Mok in the unitary case), which concerns the structure of L-packets in these cases. So our goal in this paper is to make more explicit the connection between these two very different types of theories. To do so, we reprove the results of Mœglin and Tadić in the case of quasisplit symplectic groups and orthogonal groups by using Arthur’s theory.

No abstract uploaded!

We show that the product in the quantum K-ring of a generalized flag manifold G/P involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the J-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.