Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions. Issue 1 (21st November 2019) - Shiraishi, Jun'ichi
Abstract: Based on the screened vertex operators associated with the affine screening operators, we introduce the formal power series f^gl_N(x, p|s, κ|q, t) which we call the non-stationary Ruijsenaars function . We identify it with the generating function for the Euler characteristics of the affine Laumon spaces. When the parameters s and κ are suitably chosen, the limit t→ q of f^gl_N(x, p|s, κ|q, q/t) gives us the dominant integrable characters of sl_N multiplied by 1/(p^N;p^N)_∞ ( i.e. the gl_1 character). Several conjectures are presented for f^gl_N(x, p|s, κ|q, t), including the bispectral and the Poincaré dualities, and the evaluation formula. The main conjecture asserts that (i) one can normalize f^gl_N(x, p|s, κ|q, t) in such a way that the limit κ→ 1 exists, and (ii) the limit f^st. gl_N(x, p|s|q, t) gives us the eigenfunction of the elliptic Ruijsenaars operator. The non-stationary affine q -difference Toda operator T^gl_N(κ) is introduced, which comes as an outcome of the study of the Poincaré duality conjecture in the affine Toda limit t→ 0 . The main conjecture is examined also in the limiting cases of the affine q -difference Toda (t→ 0 ), and the elliptic Calogero–Sutherland (q, t→ 1 ) equations.