#### Alladi Ramakrishnan Hall

#### Demazure flags, Chebyshev polynomials, Mock theta functions.

#### Rekha Biswal (PhD thesis defence)

##### IMSc

*The g[t]-stable Demazure modules are of great interest because of*

their connections to representation theory of quantum affine algebras.

These modules are indexed by a pair (\ell, \lambda) where \ell is a

positive integer and \lambda is a dominant integral weight of g and are

denoted as D(\ell, \lambda). Naoi proves that for m \geq \ell, D(\ell,

\lambda) admits a level m Demazure flag for an arbitrary simple Lie

algebra g. Chari et al. gave a direct and constructive proof of Naoi's

theorem in the case of sl_2. In this talk, we will discuss the level

m-Demazure flag of D(\ell, \lambda) for the current algebra associated to

sl_2. We will see how the generating series of numerical multiplicities

of Demazure modules in the Demazure flag of local Weyl modules relates to

Chebyshev polynomials and how the generating series of graded

multiplicities of Demazure modules in local Weyl modules relates to

Ramanujan's fifth order mock theta functions (surprisingly) in certain

special cases .

Done