The q-characters at roots of unity

We introduce the notion of a spectral character for finite-dimensional representations of affine algebras. These can be viewed as a suitable q=1 limit of the elliptic characters defined by Etingof and Moura for quantum affine algebras. We show that these characters determine blocks of the category of finite-dimensional modules for affine algebras. To do this we use the Weyl modules defined by Chari and Pressley and some indecomposable reducible quotient of the Weyl modules.

We resolve a long-standing puzzle in the theory of $q$-characters of finite-dimensional representations of the quantum affine algebra $U_q(\widehat{\mathfrak g})$: the seeming absence of a $q$-analogue of the Weyl group symmetry of characters of finite-dimensional representations of ${\mathfrak g}$. Namely, we define an action of the Weyl group $W$, but not on the ring ${\mathcal Y}$ of Laurent polynomials where the $q$-character homomorphism takes values. Rather, we define i...

Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases.

We give a set of sufficient conditions for a Laurent polynomial to be the q-character of a finite-dimensional irreducible representation of a quantum affine group. We use this result to obtain an explicit path description of q-characters for a class of modules in type B. In particular, this proves a conjecture of Kuniba-Ohta-Suzuki.

We describe the underlying U_q(g)--module structure of representations of quantum affine algebras.

Generalizing our earlier work, we introduce the homogeneous quantum $Z$-algebras for all quantum affine algebras $\alg$ of type one. With the new algebras we unite previously scattered realizations of quantum affine algebras in various cases. As a result we find a realization of $U_q(F_4^{(1)})$.

We use the theory of q-characters to establish a number of short exact sequences in the category of finite-dimensional representations of the quantum affine groups of types A and B. That allows us to introduce a set of 3-term recurrence relations which contains the celebrated T-system as a special case.

In this survey article for the Encyclopedia of Mathematical Physics, 2nd Edition, I give an introduction to quantum character varieties and quantum character stacks, with an emphasis on the unification between four different approaches to their construction.

The so called quantized algebras of functions on affine Hecke algebras of type A and the corresponding q-Schur algebras are defined and their irreducible unitarizable representations are classified.

We classify the finite dimensional representations of the double affine Hecke algebra of type $C^{\vee}C_1$ in the case when $q$ is not a root of 1.