solidot新版网站常见问题,请点击这里查看。

Duality for finite Gelfand pairs. (arXiv:1707.03862v1 [math.RT])

来源于:arXiv
Let $\mathrm{G}$ be a split reductive group, $K$ be a non-Archimedean local field, and $O$ be its ring of integers. Satake isomorphism identifies the algebra of compactly supported invariants $\mathbb{C}_c[\mathrm{G}(K)/\mathrm{G}(O))]^{\mathrm{G}(O)}$ with a complexification of the algebra of characters of finite-dimensional representations $\mathcal{O}(\mathrm{G}^L(\mathbb{C}))^{\mathrm{G}^L(\mathbb{C})}$ of the Langlands dual group. In this note we report on the results of the study of analogues of such an isomorphism for finite groups. In our setup we replaced Gelfand pair $\mathrm{G}(O)\subset \mathrm{G}(K)$ by a finite pair $H\subset G$. It is convenient to rewrite the character side of the isomorphism as $\mathcal{O}(\mathrm{G}^L(\mathbb{C}))^{\mathrm{G}^L(\mathbb{C})}=\mathcal{O}((\mathrm{G}^L(\mathbb{C})\times \mathrm{G}^L(\mathbb{C}))/\mathrm{G}^L(\mathbb{C}))^{\mathrm{G}^L(\mathbb{C})}$. We replace diagonal Gelfand pair $\mathrm{G}^L(\mathbb{C})\subset \mathrm{G}^L(\mathbb{C 查看全文>>