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

If one restricts an irreducible representation VλVλ of GL2nGL2n to the orthogonal group (respectively the symplectic group), the trivial representation appears with multiplicity one if and only if all parts of λ   are even (resp. the conjugate partition λ′λ′ is even). One can rephrase this statement as an integral identity involving Schur functions, the corresponding characters. Rains and Vazirani considered q,tq,t-generalizations of such integral identities, and proved them using affine Hecke algebra techniques. In a recent paper, we investigated the q=0q=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.