ScalingStacks

We let \(k\) denote the rationals \(\mathbb{Q}\) or, more generally, a commutative \(\mathbb{Q}\)-algebra. We consider the \(k\)-linear monoidal categories \(\mathrm{Sbim}_n\) of Soergel bimodules for the symmetric group \(S_n\) acting on its natural representation. In this section, if not specified otherwise, categories mean ordinary categories (in contrast to \(\infty\)-categories used later) and functors mean ordinary functors. For a fixed nonnegative integer \(n\), let \(R_n=k[x_1,x_2,\ldots, x_n]\) denote the polynomial ring over \(k\) in \(n\) variables viewed as polynomial functions on \(\mathfrak{h}^*=(k^n)^*\) in the standard way. Permuting the basis vectors of \(k^n\) induces a left action of the symmetric group \(W=S_n\) on \(R_n\) such that the simple transposition \(s_i=(i,i+1)\) acts by swapping the variables \(x_i\) and \(x_{i+1}\). Denote \(\check\alpha_i=x_{i}-x_{i+1}\) for \(1\leq i\leq n-1\). Then restriction to the span of the \(\check\alpha_i\)’s gives the usual geometric representation of \(W\) viewed as the Coxeter group generated by the simple transpositions. For any subgroup \(G\) of \(W\) let \(R_n^G\) be the subalgebra of \(G\)-invariants in \(R_n\). In case \(G=\langle s_i\rangle\) for some \(1\leq i\leq n-1\) we abbreviate \(R_n^G=R_n^i\). We will view \(R_n\) as a graded (by which we mean \(\mathbb{Z}\)-graded) algebra by putting the generators \(x_i\) in degree \(2\). Note that \(R_n^i\) is a graded subalgebra and we have a canonical, grading-preserving decomposition \[ R_n= R_n^i\oplus \check\alpha_iR_n^i\simeq R_n^i\oplus R_n^i\langle 2\rangle\] as graded \(R_n^i\)-bimodules. Here and in the following we denote for \(j\in \mathbb{Z}\) and a graded (bi)module \(M=\oplus_{i\in\mathbb{Z}} M_i\) by \(M\langle j\rangle\) the graded (bi)module which equals \(M\) as (bi)module but with the grading shifted up by \(j\), i.e. \(M\langle j\rangle_i=M_{i-j}\). The grading shifting functors \(\langle j\rangle\), \(j\in\mathbb{Z}\) equip the category of graded \((R_n,R_m)\)-bimodules for fixed \(n,m\) with an action of the group \(\mathbb{Z}\).