  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.
