# Mathematics Colloquia and Seminars

### A $p$-adic interpretation of some integral identities for Hall-Littlewood polynomials
If one restricts an irreducible representation of $GL_{n}$ to the orthogonal subgroup (respectively, the symplectic subgroup), classical branching rules tell us when the trivial representation is contained in the restricted representation. In both cases, the partition $\lambda$ that indexes the original representation must satisfy a particular condition: in the orthogonal (respectively, symplectic) case, $\lambda$ (resp. $\lambda'$) must have all even parts. Using character theory, these results may be rephrased in terms of integrals involving the Schur functions. Since Hall-Littlewood polynomials are $t$-generalizations of Schur functions, one may consider $t$-analogs of these results. We will discuss these identities, focusing on an interpretation using $p$-adic representation theory that parallels the Schur case.