Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Duality for finite Gelfand pairs   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 $\mathrm{G}^L(\mathbb{C})\subset
\mathrm{G}^L(\mathbb{C})\times \mathrm{G}^L(\mathbb{C})$ by a dual finite pair
$\check{H}\subset \check{G}$ and use Satake isomorphism as a defining property
of the duality. In this text we make a preliminary study of such duality and
compute a number of nontrivial examples of dual pairs $(H,G)$ and $(\check{H},
\check{G})$. We discuss a possible relation of our constructions to String
Topology.

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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: math-ph