A p-adic interpretation of some integral identities for Hall-Littlewood polynomials

If one restricts an irreducible representation $V_{\lambda}$ of $Gl_{2n}$ to the orthogonal group (respectively the symplectic group), the trivial representation appears with multiplicity one if and only if all parts of $\lambda$ are even (resp. the conjugate partition $\lambda'$ is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered $q,t$-generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the $q=0$ limit (Hall-Littlewood), and provided direct combinatorial arguments for these identities; this approach led to various generalizations and a finite-dimensional analog of a recent summation identity of Warnaar. In this paper, we reformulate some of these results using $p$-adic representation theory; this parallels the representation-theoretic interpretation in the Schur case. The nonzero values of the identities are interpreted as certain $p$-adic measure counts. This approach provides a $p$-adic interpretation of these identities (and a new identity), as well as independent proofs. As an application, we obtain a new Littlewood summation identity that generalizes a classical result due to Littlewood and Macdonald. Finally, our $p$-adic method also leads to a generalized integral identity in terms of Littlewood-Richardson coefficients and Hall polynomials.