The deformed Lefschetz operator L_zeta on B_x B_s

The operator. Fix $x\in S_n$, a simple reflection $s$, a strictly dominant $\rho \in S_2$ (The reduced Soergel module and the dominant Lefschetz operator) and a real parameter $\zeta \ge 0$. On the tensor product $B_x^R{B}_s^R=B_x^R{\otimes }_S{B}_s^R$ define $L_{\zeta }:=(\rho \cdot (-))\otimes {\mathrm{i}\mathrm{d}}_{B_s^R}+{\mathrm{i}\mathrm{d}}_{B_x^R}\otimes (\zeta \rho \cdot (-)),$ where the first summand is left multiplication by $\rho$ on the first factor and the second summand is left multiplication by $\zeta \rho$ on the second factor. Because the polynomial ring is commutative and the balanced tensor product identifies $br\otimes c=b\otimes rc$ for $r\in S$, the second summand equals right multiplication by $\zeta \rho$ on the first factor; both summands are degree-two, $S$-linear on the right, and commute, so $L_{\zeta }$ is a degree-two right-$S$-linear endomorphism of $B_x^R{B}_s^R$. In particular $L_{\zeta }$ preserves the submodule $M{S}_{+}$ for $M=B_x^R{B}_s^R$ and therefore descends to a degree-two $\mathbb{R}$-linear endomorphism of the reduced module $\overline{B_x^R{B}_s^R}$ (The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction supplies the induced invariant form and the reduction conventions of The reduced Soergel module and the dominant Lefschetz operator apply).

Symmetry. Each summand of $L_{\zeta }$ is multiplication by an element in the commutative position algebra of the tensor product: the first is multiplication by $\rho$ on the first factor, the second is right (equivalently middle) multiplication by $\zeta \rho$, which is self-adjoint for the induced invariant form by the invariance identities of the form. Hence $L_{\zeta }$ is symmetric for the induced form, and the Lefschetz forms $(,{)}_{L_{\zeta }}^{-i}$ and the primitive subspaces $P_{L_{\zeta }}^{-i}$ are those of Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR for this operator.

The deformation statements. Write $\mathrm{h}\mathrm{L}(x,s{)}_{\zeta }$, $\mathrm{H}\mathrm{R}(x,s{)}_{\zeta }$ and $\mathrm{H}\mathrm{R}\mathrm{b}\mathrm{a}\mathrm{r}(x,s{)}_{\zeta }$ for centered hard Lefschetz, the Hodge-Riemann relations and the strong embedded-form statement on $\overline{B_x^R{B}_s^R}$ for the operator $L_{\zeta }$, with the bottom degree $-\mathrm{\ell }(x)-1$ and the standard sign $(-1{)}^{(\mathrm{\ell }(x)+1-i)\mathrm{/}2}$ of the dictionary in Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR; the definitions of $\mathrm{H}\mathrm{R}(x,s{)}_{\zeta }$ and $\mathrm{H}\mathrm{R}\mathrm{b}\mathrm{a}\mathrm{r}(x,s{)}_{\zeta }$ refer to a fixed reduced-word induced form respectively to all choices of such forms. At $\zeta =0$ the operator $L_0=\rho \cdot (-)\otimes \mathrm{i}\mathrm{d}$ is left multiplication by $\rho$ on the first factor, so the zero-parameter statements $\mathrm{h}\mathrm{L}(x,s{)}_0$, $\mathrm{H}\mathrm{R}(x,s{)}_0$, $\mathrm{H}\mathrm{R}\mathrm{b}\mathrm{a}\mathrm{r}(x,s{)}_0$ are exactly the undeformed statements $\mathrm{h}\mathrm{L}(x,s)$, $\mathrm{H}\mathrm{R}(x,s)$, $\mathrm{H}\mathrm{R}\mathrm{b}\mathrm{a}\mathrm{r}(x,s)$ of the dictionary. For fixed data the family $\zeta \mapsto L_{\zeta }$ is polynomial of degree one in $\zeta$.

Summand stability. Any direct summand $B\subseteq B_x^R{B}_s^R$ that is a sub-bimodule is stable under $L_0$, because left multiplication by the central element $\rho$ is a bimodule endomorphism; stability under $L_{\zeta }$ for $\zeta >0$ is a separate condition, since a projector onto $B$ need not commute with the middle-position operator. The undeformed summand implications used in the induction of this page are supplied by the local restriction lemma later on this page, after deforming the parameter back to $\zeta =0$; no statement of this definition identifies a restricted positive-parameter operator with a restriction of $L_{\zeta }$.