Hall-Littlewood多项式某些积分恒等式的p-adic解释
A p-adic interpretation of some integral identities for Hall-Littlewood polynomials
摘要：
如果将$Gl_{2n}$的不可约表示$V_λ$限制到正交群（相应地辛群），则平凡表示出现重数为1当且仅当$λ$的所有部分都是偶数（相应地，共轭分配$λ'$是偶数）。我们可以把这句话改写成一个包含舒尔函数和相应特征的积分恒等式。Rains和Vazirani考虑了这种积分恒等式的$q,t$ -推广，并使用仿射Hecke代数技巧证明了它们。在最近的一篇论文中，我们研究了$q=0$极限（Hall-Littlewood），并为这些恒等式提供了直接的组合参数;这种方法导致了各种推广和Warnaar最近求和恒等式的有限维模拟。在本文中，我们重新制定了这些结果中的一些使用$p$ -进表示论，这平行的Schur情况下的表示理论的解释。恒等式的非零值被解释为某些$p$ -adic测量计数。这种方法提供了一个$p$ -adic解释这些身份（和一个新的身份），以及独立的证明。作为应用，我们得到了一个新的Littlewood求和恒等式，它推广了Littlewood和Macdonald的一个经典结果。最后，我们的$p$ -adic方法也导致了一个广义积分恒等式的Littlewood-Richardson系数和霍尔多项式。
英文摘要：
If one restricts an irreducible representation $V_λ$ 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 $λ$ are even (resp. the conjugate partition $λ'$ 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.
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