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Register a new userPeter-Weyl, Howe and Schur-Weyl theorems for current groups
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The classical Peter-Weyl theorem describes the structure of the space of functions on a semi-simple algebraic group. On the level of characters (in type A) this boils down to the Cauchy identity for the products of Schur polynomials. We formulate and prove the analogue of the Peter-Weyl theorem for the current groups. In particular, in type A the corresponding characters identity is governed by the Cauchy identity for the products of q-Whittaker functions. We also formulate and prove a version of the Schur-Weyl theorem for current groups. The link between the Peter-Weyl and Schur-Weyl theorems is provided by the (current version of) Howe duality.
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We study a generalization of the results in cite{cfk} to the case of $SU(1|1)$ interpreted as the supercircle $S^{1|2}$. We describe all of its finite dimensional complex irreducible representations, we give a reducibility result for representations not containing the trivial character, and we compute explicitly the corresponding matrix elements. In the end we give the Peter-Weyl theorem for $S^{1|2}$.
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