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 $f^{\widehat{\mathfrak gl}_N}(x, p|s, κ|q, t)$ 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 $f^{\widehat{\mathfrak gl}_N}(x, p|s, κ|q, q/t)$ gives us the dominant integrable characters of $\widehat{\mathfrak sl}_N$ multiplied by $1/(p^N;p^N)_∞$ ( i.e. the $\widehat{\mathfrak gl}_1$ character). Several conjectures are presented for $f^{\widehat{\mathfrak gl}_N}(x, p|s, κ|q, t)$, including the bispectral and the Poincaré dualities, and the evaluation formula. The main conjecture asserts that (i) one can normalize $f^{\widehat{\mathfrak gl}_N}(x, p|s, κ|q, t)$ in such a way that the limit $κ→ 1$ exists, and (ii) the limit $f^{{\rm st.}\, \widehat{\mathfrak gl}_N}(x, p|s|q, t)$ gives us the eigenfunction of the elliptic Ruijsenaars operator. The non-stationary affine $q$ -difference Toda operator ${\mathcal T}^{\widehat{\mathfrak gl}_N}(κ)$ 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.