Metaplectic Iwahori Whittaker functions and supersymmetric lattice models

Abstract

In this paper we consider Iwahori Whittaker functions on n-fold metaplectic covers G of G(F ) with G a split reductive group over a non-archimedean local field F . For every element φ of a basis of Iwahori Whittaker functions, and for every g ∈ G, we evaluate φ(g) by recurrence relations over the Weyl group using "vector Demazure-Whittaker operators." Specializing to the case of G = GL r , we exhibit a solvable lattice model whose partition function equals φ(g). These models are of a new type associated with the quantum affine super group U q ( gl(r|n)). The recurrence relations on the representation theory side then correspond to solutions to Yang-Baxter equations for the lattice models.

Iwahori-metaplectic duality

“…We will explain how in Section 4. These recurrences, and even the base case are different from the known Demazure recurrences for the r!n r Iwahori metaplectic Whittaker functions ( [12,25,4]). Moreover, Iwahori vectors and Whittaker functions have connections with Kazhdan-Lusztig theory and the geometry of Bott-Samelson resolutions ( [27,18,26,9,24]).…”

“…Instead, horizontal edges along a path were assigned elements in Z/nZ called charges which increase along the path. In [4] we developed another formulation based on attributes that are constant along the path. In fact, the models in [4] compute the more general metaplectic Iwahori Whittaker functions (and not only spherical) by introducing two sets of paths: one set taking colors from an ordered palette P r describing the Iwahori data as in Section 3.2.1 and the other set taking colors from another ordered palette Pn describing the metaplectic data earlier captured by the charge attributes.…”

“…In [4] we developed another formulation based on attributes that are constant along the path. In fact, the models in [4] compute the more general metaplectic Iwahori Whittaker functions (and not only spherical) by introducing two sets of paths: one set taking colors from an ordered palette P r describing the Iwahori data as in Section 3.2.1 and the other set taking colors from another ordered palette Pn describing the metaplectic data earlier captured by the charge attributes. We gave the elements of the second palette Pn the name supercolors because of how these connect to the odd part of the super quantum group U q ( gl(r|n)) associated to the lattice model.…”

“…Besides using the notation w i instead of ci in [4], the supercolored paths were there also left-moving, exiting at the left boundary, but this is not a feature of supercolored paths per se, but rather a feature of the model being a generalization of Gamma metaplectic ice instead of Delta metaplectic ice which we will consider here, and where the supercolored paths will be right moving.…”

“…Our initial inspiration came from [4], where we give lattice models whose partition functions are Iwahori Whittaker functions on the metaplectic groups, thereby unifying the two cases described above using the associated quantum group U q ( gl(r|n)). But perhaps even more importantly, the paper [4] treated the two specializations to the non-metaplectic Iwahori case and to the spherical metaplectic case on similar footing, as evoked by the terminology for the spins in terms of colors and supercolors ("scolors"). The Iwahori case amounts to restricting to only colors, while the metaplectic case amounts to restricting to only supercolors.…”