We relate Iwahori-Whittaker functions on metaplectic covers to certain Demazure-Lusztig operators, the latter of which are built from a Weyl group action previously considered by G. Chinta and P. Gunnells. Using a certain combinatorial identity for the sum of these Demazure-Lusztig operators, we obtain an analogue of the Casselman-Shalika formula for spherical Whittaker functions in this context.

In [CG10] the first two named authors defined an action of a Weyl group on rational functions and used it to construct multiple Dirichlet series. These series are related to Whittaker functions on an n-fold metaplectic cover of a reductive group. In this paper, we define metaplectic analogues of the Demazure and Demazure-Lusztig operators. We show how these operators can be used to recover the formulas from [CG10], and how, together with results of McNamara [McN], they can be used to compute Whittaker functions on metaplectic groups over p-adic fields.

Starting from some linear algebraic data (a Weyl-group invariant bilinear form) and some arithmetic data (a bilinear Steinberg symbol), we construct a cover of a Kac-Moody group generalizing the work of Matsumoto. Specializing our construction over non-archimedean local fields, for each positive integer n we obtain the notion of n-fold metaplectic covers of Kac-Moody groups. In this setting, we prove a Casselman-Shalika type formula for Whittaker functions.In generalizing the above to a Kac-Moody context, we could perhaps have chosen any of the existing constructions (not all of which are equivalent) for Kac-Moody groups. We choose here the functorial approach of Tits [43] 3 , whose input is a "root datum" consisting of a quadruple (Y, {y i }, X , {x i }) where Y, X are to the play the role of the cocharacter and character lattice respectively of the group, and y i and x i the coroots and roots respectively. In this setup the "simply connected" groups will roughly be the ones for which Y is spanned by the {y i }, though some further care is required when A is not of full rank. See §2.1.6.To carry over Matsumoto's strategy, we first need to fix an integral Weyl group invariant form Q on Y. In the "simply-connected" case just described, these are again easily classified (cf. Proposition 2.2.2). Starting from the choice of such Q and a positive integer n, we can also form a "metaplectic root datum" as in [33] or [46]. We use this new root datum to describe the Casselman-Shalika formula and the metaplectic Demazure-Lusztig operators, but we do not actually need the construction of the metaplectic dual group. In fact, our restrictions on the cardinality of the residue field mask almost all of the subtleties of this dual group which Weissman has constructed in [46]. In the affine setting, we tabulate the possibilities for metaplectic dual root systems in Table 2.3.2. Fixing Q and choosing some Steinberg symbol, Matsumoto's strategy carries over with little change to the Kac-Moody context. KM1. The construction of the cover of the torus and its family of automorphisms follows as in [29]. The possible complications involving non-degeneracy of the Cartan matrix (e.g. the loop rotation in the affine case) are conveniently taken care of using the "Q-formalism." KM2. The next step is to obtain a presentation for a group N that plays a role analogous to the normalizer of the torus in the finite-dimensional context, and then refine this to a presentation of the "integral" version of this normalizer. We follow the classical arguments of Tits here [41] adapted with almost no change to the Kac-Moody context. Here we use the simply-connected assumption to obtain a simple presentation for N, though it should be possible to remove this assumption. KM3. Finally, the construction of a cover E as a group of operators on a set S (again constructed as a fiber product using the cover of N and the Bruhat decompostion) follows as in Matsumoto. Recall also that Matsumoto's argument involves a rank two check, and at first glance it might seem...

In this paper we study the security of a proposal for Post-Quantum Cryptography from both a number theoretic and cryptographic perspective. Charles-Goren-Lauter in 2006 [CGL06] proposed two hash functions based on the hardness of finding paths in Ramanujan graphs. One is based on Lubotzky-Phillips-Sarnak (LPS) graphs and the other one is based on Supersingular Isogeny Graphs. A 2008 paper by Petit-Lauter-Quisquater breaks the hash function based on LPS graphs. On the Supersingular Isogeny Graphs proposal, recent work has continued to build cryptographic applications on the hardness of finding isogenies between supersingular elliptic curves. A 2011 paper by De Feo-Jao-Plût proposed a cryptographic system based on Supersingular Isogeny Diffie-Hellman as well as a set of five hard problems. In this paper we show that the security of the SIDH proposal relies on the hardness of the SSIG path-finding problem introduced in [CGL06]. In addition, similarities between the number theoretic ingredients in the LPS and Pizer constructions suggest that the hardness of the path-finding problem in the two graphs may be linked. By viewing both graphs from a number theoretic perspective, we identify the similarities and differences between the Pizer and LPS graphs.

The polarizations of one relation of degree five and two relations of degree six minimally generate the ideal of relations among a minimal generating system of the algebra of multisymmetric polynomials in an arbitrary number of three-dimensional vector variables. In the general case of n-dimensional vector variables, a relation of degree 2n among the polarized power sums is presented such that it is not contained in the ideal generated by lower degree relations.MSC: 13A50, 14L30, 20G05