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If one restricts an irreducible representation $V_{\lambda}$ of $Gl_{2n}$ tothe orthogonal group (respectively the symplectic group), the trivialrepresentation appears with multiplicity one if and only if all parts of$\lambda$ are even (resp. the conjugate partition $\lambda'$ is even). One canrephrase this statement as an integral identity involving Schur functions, thecorresponding characters. Rains and Vazirani considered $q,t$-generalizationsof such integral identities, and proved them using affine Hecke algebratechniques. In a recent paper, we investigated the $q=0$ limit(Hall-Littlewood), and provided direct combinatorial arguments for theseidentities; this approach led to various generalizations and afinite-dimensional analog of a recent summation identity of Warnaar. In thispaper, we reformulate some of these results using $p$-adic representationtheory; this parallels the representation-theoretic interpretation in the Schurcase. The nonzero values of the identities are interpreted as certain $p$-adicmeasure counts. This approach provides a $p$-adic interpretation of theseidentities (and a new identity), as well as independent proofs. As anapplication, we obtain a new Littlewood summation identity that generalizes aclassical result due to Littlewood and Macdonald. Finally, our $p$-adic methodalso leads to a generalized integral identity in terms of Littlewood-Richardsoncoefficients and Hall polynomials.

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