List of papers by A. Kirillov, Jr

• Representations of affine Lie algebras, parabolic differential equations and Lame functions, with Pavel Etingof. Published in Duke Math. J., vol. 74(1994), pp. 585--614.
• A unified representation-theoretic approach to special functions, with Pavel Etingof. This is a short (4 pages) note in which we sketch the program of special functions related with intertwining operators. Also available in Russian. Published (in Russian) in Funct. Analysis and its Applic., vol. 28(1994), pp. 91--94.
• Macdonald's polynomials and representations of quantum groups, with Pavel Etingof. In this paper we show how one can obtain Macdonald's polynomilas of type A from intertwining operators for sl(n). Published in Math. Res. Let., vol 1(1994), pp. 279--296.
• On the affine analogue of Jack and Macdonald's polynomials, with Pavel Etingof Published in Duke Math. J., vol. 78(1995), pp. 229--256.
• A representation-theoretic proof of inner product and symmetry identities for Macdonald's polynomials, with Pavel Etingof. Published in Compositio Math. 102 (1996), no. 2, pp. 179--202.
• Lectures on the affine Hecke algebras and Macdonald conjectures. These are notes of a lecture course devoted to Cherednik's proof of Macdonald's inner product conjectures by means of Hecke algebras. Published in Bull. Amer. Math. Soc. 34 (1997), pp. 251--292.
• Spherical functions on affine Lie groups, with P. Etingof and I. Frenkel. Published in Duke Math. J., vol. 80 (1995), pp. 59--90.
• Traces of intertwining operators and Macdonald's polynomials.
  This is my Ph.D. dissertation, which summarizes the results of the above series of papers on special functions related with intertwining operators, among them - Macdonald's polynomials, correlation functions in WZW model of Conformal Field Theory, etc.
• On inner product in modular tensor categories. I. J. of Amer. Math. Soc. vol. 9 (1996), 1135-1169.
  In this paper we study some properties of morphisms in modular tensor categories; in particular, we show that the action of modular group on the morphisms is unitary with respect to a natural inner product. We also show that in the case of category coming from representations of Uqsl(n) at roots of unity many (for sl(2) -all) of matrix elements of S-matrix can be written in terms of special values of Macdonald's polynomials, which gives new identities for these polynomials.
• On inner product in modular tensor categories II. Inner product on conformal blocks and affine inner product identities. Adv. Theor. Math. Phys. vol 2 (1998), 155-180.
  This is a continuation of the previous paper. Here we apply the same constructions as before to the MTC coming from the integrable representations of affine Lie algebras. In this case our construction immediately gives a hermitian form on the spaces of conformal blocks, and this form is modular invariant (Warning: we cannot prove that it is positive definite). We show that this form can be rewritten in terms of asymptotics of KZ equations, and calculate it for sl(2), in which case the formula is a natural affine analogue of Macdonald's inner product identities. We also formulate as a conjecture similar formula for sl(n).
• Canonical basis and homology of local systems, with Igor Frenkel and Alexander Varchenko. In this paper, we use the isomorphism between highest weight U_q(sl(2))-modules and homologies of certain local systems on the configuration spaces, constructed by Varchenko, to give a geometric construction of the dual of the Lusztig's canonical basis in a tensor product of irreducible finite-dimensional U_q(sl(2))-modules. Published in IMRN, vol. 16 (1997), pp. 783--806.
• Kazhdan-Lusztig polynomials and canonical basis, with I. Frenkel and M. Khovanov. To appear in Transformation Groups. ( here is the style file of this journal, necessary to TeX this paper).
• On Cherednik-Macdonald-Mehta identities, with Pavel Etingof.
• Lectures on Representation Theory and Knizhnik-Zamolodchikov equations, AMS, 1998.
  A 200-page book: Igor Frenkel's lecture course written down and revised by Igor himself, Pasha Etingof, and myself.