The reduced Soergel module and the dominant Lefschetz operator

The reduction. Work over the ring $S=\mathbb{R}[x_1,\dots ,x_n]$ of the real scalar-extension definition (The real type-A Soergel category by scalar extension), with $S_{+}={⨁}_{d>0}{S}_d$ the ideal of positive-degree elements; the residue field of the right action is $k:=S\mathrm{/}{S}_{+}=\mathbb{R},$ with quotient map $S\twoheadrightarrow k$, $s\mapsto \bar{s}$. For a graded $(S,S)$-bimodule $M$ that is finite free as a right $S$-module put $\bar{M}:=M{\otimes }_{S}k=M\mathrm{/}M{S}_{+},$ the reduction of $M$: a finite-dimensional graded $k$-vector space, on which the left $S$-action descends because $S_{+}\subseteq S$ is central and $M{S}_{+}$ is stable under left multiplication. Write ${\bar{M}}^i$ for its homogeneous component of degree $i$, $d_i(M):={\dim }_{\mathbb{R}}{\bar{M}}^i$ for the graded dimensions, and $\dim \bar{M}:={\sum }_{i\in \mathbb{Z}}{d}_i(M)v^{-i}$ for the graded dimension (a finite Laurent polynomial); all tensor products below are over $S$. For a morphism $f:M\to M^{\mathrm{\prime }}$ of graded bimodules, i.e. a degree-zero bi-$S$-linear map, the induced map $\bar{f}:\bar{M}\to \overline{M^{\mathrm{\prime }}},\bar{f}(m+M{S}_{+}):=f(m)+M^{\mathrm{\prime }}{S}_{+},$ is well defined and $k$-linear of degree zero, because $f(M{S}_{+})\subseteq M^{\mathrm{\prime }}{S}_{+}$. Reduction is a functor from finite right-free graded bimodules to finite dimensional graded $k$-vector spaces; it preserves finite direct sums and shifts, in the sense $\overline{M(d)}=\bar{M}(d)$ in the Elias-Williamson shift convention $M(d{)}_e=M_{d+e}$.

The dominant Lefschetz operator. Fix a strictly dominant degree-two element $\rho ={\sum }_{j=1}^n{c}_j{x}_j\in S_2,c_1>c_2>\cdots >c_n,$ which exists by The real type-A Soergel category by scalar extension (for example $c_j=(n+1)\mathrm{/}2-j$); recall that this is exactly the condition $\rho ({\alpha }_i^{\vee })>0$ for every simple coroot ${\alpha }_i^{\vee }$ in the coordinate normalization of this page (Finite Weyl root system, lattice and chamber conventions). Let $\rho \cdot (-):M\to M$, $m\mapsto \rho m$, denote left multiplication by $\rho$. Since $S_{+}\subseteq S$ is central and $\rho (M{S}_{+})\subseteq M{S}_{+}$, this operator preserves the submodule $M{S}_{+}$ and therefore descends to a degree-two $k$-linear endomorphism $L_0:=\rho \cdot (-):\bar{M}⟶\bar{M},m+M{S}_{+}⟼\rho m+M{S}_{+},$ called the dominant Lefschetz operator on $\bar{M}$. For rational objects $B_x,B_s$ with real scalar extensions $B_x^R=B_x{\otimes }_{\mathbb{Q}}\mathbb{R}$ and $B_s^R$, the same construction applies to the reduced tensor product $\overline{B_x^R{B}_s^R}=\overline{(B_x{B}_s{)}^R}$, where $B_x{B}_s$ denotes the tensor product over the rational polynomial ring; the operator $L_0$ on it is the value $\zeta =0$ of the deformed operator $L_{\zeta }$ defined later on this page.

Descent detection. For every $w\in S_n$ and every simple reflection $s_i$ one has $(w\rho )({\alpha }_i^{\vee })>0⟺\mathrm{\ell }(s_{i}w)>\mathrm{\ell }(w)\mathrm{.}$ Indeed, $(w\rho )({\alpha }_i^{\vee })=c_{w^{-1}(i)}-c_{w^{-1}(i+1)}$ is positive exactly when $w^{-1}(i)<w^{-1}(i+1)$, the left-ascent criterion for inversion length. Thus the displayed evaluation detects left descents; applied to $w=x^{-1}$ it detects right descents of $x$, because $\mathrm{\ell }(s_i{x}^{-1})=\mathrm{\ell }(x{s}_i)$.

Grading conventions. The reduced module $\overline{B_x^R}$ has bottom degree $-\mathrm{\ell }(x)$ and top degree $+\mathrm{\ell }(x)$; the unreduced bimodule has nonzero components in arbitrarily high degrees. On a reduced module the "centered" degree $-i$ is the internal degree $-i$, and the Lefschetz operator raises internal degree by two. All homological (complex) shifts are written $[j]$ and all internal shifts $(d)$, as on the rest of this page; the Bruhat order used throughout is the rank-inequality order of The Bruhat order on $S_n$ by rank inequalities.

The descent criterion. Write $\rho ={\sum }_j{c}_j{x}_j$; for a fixed $i$, $(w\rho )({\alpha }_i^{\vee })=c_{w^{-1}(i)}-c_{w^{-1}(i+1)}$, because $w$ preserves the standard form and $w^{-1}(x_i-x_{i+1})=x_{w^{-1}(i)}-x_{w^{-1}(i+1)}$. Hence $(w\rho )({\alpha }_i^{\vee })>0$ if and only if $w^{-1}(i)<w^{-1}(i+1)$. By the deletion criterion of Finite Weyl strong exchange and deletion, applied to $x=w^{-1}$ and the positive root ${\beta }_i=x_i-x_{i+1}$ of the positive system $\{x_a-x_b:a<b\}$ fixed on this page, one has $\mathrm{\ell }(w^{-1}{s}_i)=\mathrm{\ell }(w)-1$ if and only if $w^{-1}{\beta }_i<0$, i.e. if and only if $w^{-1}(i)>w^{-1}(i+1)$; taking inverses, $\mathrm{\ell }(s_{i}w)=\mathrm{\ell }(w)-1$ if and only if $w^{-1}(i)>w^{-1}(i+1)$. Therefore $(w\rho )({\alpha }_i^{\vee })>0$ exactly when $\mathrm{\ell }(s_{i}w)=\mathrm{\ell }(w)+1$, as claimed. No alternating-sign coroot and no chamber other than the coordinate dominant chamber is used. The instance $w=x^{-1}$ gives that $(x^{-1}\rho )({\alpha }_s^{\vee })>0$ for a right-descent test: the length $\mathrm{\ell }(s{x}^{-1})=\mathrm{\ell }(xs)$ is larger than $\mathrm{\ell }(x^{-1})=\mathrm{\ell }(x)$ precisely when $xs>x$.

The reduced length bounds and the one-dimensional extremal degrees of normalized indecomposables are proved in Normalized type-A Soergel bimodules are self-dual and have length degree bounds, from top-support classification and perfect word duality, without a character or Hodge assumption.