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## A path model for Whittaker vectors

Philippe Di Francesco
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R. Kedem
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B. Turmunkh
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#### Abstract

In this paper we construct weighted path models to compute Whittaker vectors in the completion of Verma modules, as well as Whittaker functions of fundamental type, for all finite-dimensional simple Lie algebras, affine Lie algebras, and the quantum algebra $U_q(\mathfrak{sl}_{r+1})$. This leads to series expressions for the Whittaker functions. We show how this construction leads directly to the quantum Toda equations satisfied by these functions, and to the $q$-difference equations in the quantum case. We investigate the critical limit of affine Whittaker functions computed in this way.

#### Domains

Physics [physics] Mathematical Physics [math-ph]

### Dates and versions

cea-01251620 , version 1 (06-01-2016)

### Identifiers

• HAL Id : cea-01251620 , version 1
• ARXIV :

### Cite

Philippe Di Francesco, R. Kedem, B. Turmunkh. A path model for Whittaker vectors. 2016. ⟨cea-01251620⟩

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