The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction

The Bott-Samelson bimodule and its subexpression basis. Fix a finite word $\underset{‾}{x}=s_1\cdots s_m$ in the simple reflections and let $\mathrm{B}\mathrm{S}(\underset{‾}{x}):=B_{s_1}{\otimes }_S\cdots {\otimes }_S{B}_{s_m}$ be the real Bott-Samelson bimodule (The Bott–Samelson bimodule of a word), with $B_s=S{\otimes }_{S^s}S(1)$ (The Soergel bimodule $B_i$ of a simple reflection). In each factor put $c_{\mathrm{i}\mathrm{d}}:=1\otimes 1,c_s:=\frac{1}{2}({\alpha }_s\otimes 1+1\otimes {\alpha }_s),$ of degrees $-1$ and $+1$; here ${\alpha }_s$ is the coordinate root of The real type-A Soergel category by scalar extension, so that ${\mathrm{\partial }}_s(r)=(r-sr)\mathrm{/}{\alpha }_s$ is the coordinate Demazure operator. The elements $\{c_{\mathrm{i}\mathrm{d}},c_s\}$ form a right (and a left) $S$-basis of $B_s$ and the multiplication rules $c_{\mathrm{i}\mathrm{d}}{c}_{\mathrm{i}\mathrm{d}}=c_{\mathrm{i}\mathrm{d}},c_{\mathrm{i}\mathrm{d}}{c}_s=c_s,c_s{c}_s=c_s{\alpha }_s$ hold. For a subexpression $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_m)$ with ${\epsilon }_i\in \{\mathrm{i}\mathrm{d},s_i\}$ put $c_{\epsilon }:=c_{{\epsilon }_1}\cdots c_{{\epsilon }_m}\in \mathrm{B}\mathrm{S}(\underset{‾}{x}),$ the tensor product of the chosen elements, and put $c_{\mathrm{t}\mathrm{o}\mathrm{p}}:=c_{s_1}\cdots c_{s_m}\mathrm{.}$ The elements $\{c_{\epsilon }\}$ form a right (and a left) $S$-basis of $\mathrm{B}\mathrm{S}(\underset{‾}{x})$; the bimodule is finite free of rank $2^m$ on each side (Soergel generators and Bott–Samelson products are finite free on both sides). Moreover $\mathrm{B}\mathrm{S}(\underset{‾}{x})\cong (S{\otimes }_{S^{s_1}}S{\otimes }_{S^{s_2}}\cdots {\otimes }_{S^{s_m}}S)(m)$, so after the shift by $(-m)$ the termwise multiplication makes $\mathrm{B}\mathrm{S}(\underset{‾}{x})(-m)$ a commutative ring with unit $c_{\mathrm{i}\mathrm{d}}\cdots c_{\mathrm{i}\mathrm{d}}$; in particular it contains the powers of any degree-two element and products of commuting elements.

The trace and the intersection form. Define the trace $\mathrm{Tr}:\mathrm{B}\mathrm{S}(\underset{‾}{x})⟶S$ to be the coefficient of $c_{\mathrm{t}\mathrm{o}\mathrm{p}}$ when an element is written in the right basis $\{c_{\epsilon }\}$, and define the intersection form by $⟨b,b^{\mathrm{\prime }}⟩:=\mathrm{Tr}(b{b}^{\mathrm{\prime }}),$ the product being taken in the commutative ring $\mathrm{B}\mathrm{S}(\underset{‾}{x})(-m)$. Then:

1. $⟨-,-⟩$ is a graded symmetric form, $S$-bilinear for the right action, and invariant in the sense $⟨rb,b^{\mathrm{\prime }}⟩=⟨b,r{b}^{\mathrm{\prime }}⟩$ and $⟨br,b^{\mathrm{\prime }}⟩=⟨b,b^{\mathrm{\prime }}r⟩=⟨b,b^{\mathrm{\prime }}⟩r$ for $r\in S$;
2. if $B$ is a direct summand of a finite direct sum of shifts of Bott-Samelson bimodules, then an invariant form on $B$ is obtained by restricting an invariant form of the ambient finite sum, and the restriction is determined up to the choices of the ambient forms;
3. for a self-dual bimodule $B$ with invariant form $⟨-,-{⟩}_B$ the induced form on $B{B}_s$ is the unique invariant form satisfying, for all $b,b^{\mathrm{\prime }}\in B$, $⟨\alpha (b),\alpha (b^{\mathrm{\prime }})⟩={\mathrm{\partial }}_s(⟨b,b^{\mathrm{\prime }}{⟩}_B),⟨\alpha (b),\beta (b^{\mathrm{\prime }})⟩=⟨b,b^{\mathrm{\prime }}{⟩}_B=⟨\beta (b),\alpha (b^{\mathrm{\prime }})⟩,⟨\beta (b),\beta (b^{\mathrm{\prime }})⟩=⟨b,b^{\mathrm{\prime }}{⟩}_B{\alpha }_s,$ where $\alpha (b):=b{c}_{\mathrm{i}\mathrm{d}}$ and $\beta (b):=b{c}_s$;
4. the intersection form on a Bott-Samelson bimodule is non-degenerate as an $S$-valued form, and for a summand $B$ of a Bott-Samelson bimodule the restricted form is non-degenerate whenever $B$ has finite free right $S$-rank and the induced reduced form on $\bar{B}$ is non-degenerate (graded Nakayama);
5. the reduction of $⟨-,-⟩$ is a graded symmetric $k$-valued form $⟨-,-{⟩}_{\bar{M}}$ on $\bar{M}$ (The reduced Soergel module and the dominant Lefschetz operator), and for a finite right-free $M$ the $S$-valued form is non-degenerate if and only if the reduced form is graded non-degenerate;
6. Bott-Samelson bimodules and the indecomposable bimodules $B_w^R$ are self-dual, so $\dim (\overline{B_w^R}{)}^{-i}=\dim (\overline{B_w^R}{)}^i$ for all $i$ (symmetric Betti numbers).

Additionally record the left/right asymmetry: the form is bilinear for the right action and satisfies the two invariance identities displayed in (1); it is not left $S$-bilinear in general, because $⟨rb,b^{\mathrm{\prime }}⟩$ is defined through the trace of a product while the two $S$-actions differ on $\mathrm{B}\mathrm{S}(\underset{‾}{x})$.

Why the clauses hold. (i) The product rules in each factor follow from $c_{\mathrm{i}\mathrm{d}}=1\otimes 1$, ${\alpha }_s\in S^s\oplus {\alpha }_s{S}^s$ and ${\mathrm{\partial }}_s({\alpha }_{s}h)=2h$ for $h\in S^s$; termwise multiplication makes $\mathrm{B}\mathrm{S}(\underset{‾}{x})(-m)$ commutative, and $\mathrm{Tr}$ is $S$-linear on the right. Symmetry of $⟨-,-⟩$ is commutativity of the product combined with symmetry of $\mathrm{Tr}$ on products; the invariance identities in (1) are the associativity of the commutative product: $\mathrm{Tr}(rb{b}^{\mathrm{\prime }})=\mathrm{Tr}(br{b}^{\mathrm{\prime }})$ and $\mathrm{Tr}(br{b}^{\mathrm{\prime }})=\mathrm{Tr}(b{b}^{\mathrm{\prime }}r)$. (ii) Restriction of the ambient form is invariant because the embedding of a summand and the ambient form are bimodule data, the precise scalar and multiplicity-space comparisons under the character hypotheses are proved in the uniqueness lemma later on this page. (iii) For (3), EW fix a basis $e_1,\dots ,e_m$ of $B$ and note that $\alpha (e_1),\dots ,\alpha (e_m),\beta (e_1),\dots ,\beta (e_m)$ is a basis of $B{B}_s$; formulas (3.8)-(3.10) fix the form on this basis and one checks that they are consistent for all $b,b^{\mathrm{\prime }}$ and left-invariant; the form is symmetric if $⟨-,-{⟩}_B$ is. (iv) For (4), with a basis $e_i$ and dual basis $e_i^{\ast }$ for the non-degenerate form on $B$, the matrix of the induced form in the bases $(\alpha (e_i),\beta (e_i))$ and $(\beta (e_i^{\ast }),\alpha (e_i^{\ast }))$ is $\left(\begin{array}{cc}{I}_m& 0\\ {\alpha }_s{I}_m& I_m\end{array}\right)$: the upper-right entry is ${\mathrm{\partial }}_s$ of the identity pairing, and ${\mathrm{\partial }}_s(1)=0$. Its determinant is $1$, so the induced form is non-degenerate (Lemma 3.8). Iterating this from the multiplication form on $S$ gives Corollary 3.9, the non-degeneracy of the intersection form on every $\mathrm{B}\mathrm{S}(\underset{‾}{x})$; the statement about summands is the graded Nakayama criterion of (5). (v) For (5), an invariant form on a finite right-free $M$ is the same as a bimodule map $M\to DM$; the reduction functor and the graded Nakayama lemma show that this map is an isomorphism if and only if its reduction is, and the reduced form is graded symmetric because it is induced by a graded form. (vi) For (6), the intersection form is non-degenerate on $\mathrm{B}\mathrm{S}(\underset{‾}{x})$, hence gives an isomorphism $\mathrm{B}\mathrm{S}(\underset{‾}{x})\to D\mathrm{B}\mathrm{S}(\underset{‾}{x})$; for the indecomposables $B_w^R$, normalized self-duality is proved directly from the unique top summand of a perfect self-dual word in Normalized type-A Soergel bimodules are self-dual and have length degree bounds; an isomorphism $B\cong DB$ induces $\bar{B}\cong (\bar{B}{)}^{\ast }$ as graded real vector spaces: for a finite homogeneous right $S$-basis, right $S$-duality reverses its generator degrees, and right reduction turns that dual basis into the real dual basis. This proves the symmetry of the reduced Betti numbers. All constructions are compatible with the dominant Lefschetz operator $L_0=\rho \cdot (-)$ of The reduced Soergel module and the dominant Lefschetz operator because the commutative ring structure is spanned by the $c_{\epsilon }$ and left multiplication by $\rho$ is self-adjoint for every invariant form.