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Register a new userComparison of the depths on both sides of the local Langlands correspondence for Weil-restricted groups (with appendix by Jessica Fintzen)
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Let $E/F$ be a finite and Galois extension of non-archimedean local fields. Let $G$ be a connected reductive group defined over $E$ and let $M: = mathfrak{R}_{E/F}, G$ be the reductive group over $F$ obtained by Weil restriction of scalars. We investigate depth, and the enhanced local Langlands correspondence, in the transition from $G(E)$ to $M(F)$. We obtain a depth-comparison formula for Weil-restricted groups.
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Let $K$ be a non-archimedean local field. In the local Langlands correspondence for tori over $K$, we prove an asymptotic result for the depths.
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A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers - connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for $SL(N)$. We introduce a difference equation version of opers called $q$-opers and prove a $q$-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted $q$-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the $q$-Langlands correspondence. We also describe an application of $q$-opers to the equivariant quantum $K$-theory of the cotangent bundles to partial flag varieties.
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