Soegel Bimodules

Soegel bimodules

Geometric origin

How to construct interesting families of bimodules out of semi-simple complexes on $G/B$?

Let $\mathscr{G}$ be a complex reductive group over $\mathbb{C}$ with Borel $\mathscr{B}$ and maximal torus $\mathscr{T}\subset \mathscr{B}$.

Let $\mathscr{X}:=\mathscr{G}/\mathscr{B}$ be the flag variety and consider $D^{b}(\mathscr{B}\backslash \mathscr{X})$ endowed with perverse t-structure. The Bruhat decomposition gives a stratification $\mathscr{X}=\sqcup_ {w\in W}\mathscr{X}_ {w}$. Simple objects in the heart $\mathrm{Perv}(\mathscr{B}\backslash \mathscr{X})$ are parameterized by $W$, i.e. $\mathrm{IC}_ {w}:=\mathrm{IC}(\mathscr{X}_ {w},\underline{\mathbb{Q}})$.

Let $\mathrm{IC}_ {\mathscr{B}}(\mathscr{X},\mathbb{Q})$ be the full subcategory of $D^{b}_ {\mathscr{B}}(\mathscr{X})$ consisting of semi-simple objects. The category $D^{b}(\mathscr{B}\backslash \mathscr{X})$ has a convolution product. By BBDG decomposition theorem, $\mathrm{IC}_ {\mathscr{B}}(\mathscr{X},\mathbb{Q})$ is closed under this convolution $\ast$.

Let $\mathbb{X}$ denote the character lattice of $\mathscr{T}$. Then consider the $\mathbb{Q}$-algebra $$R:=\mathrm{Sym}^{\ast}(\mathbb{Q}\otimes_ {\mathbb{Z}}\mathbb{X})$$ endowed with a grading such that $(\mathbb{Q}\otimes_ {\mathbb{Z}}\mathbb{X})$ is in degree 2.

Let $R\text{-}\mathrm{Mod}^{\mathbb{Z}}\text{-}R$ be the abelian category of graded $R$-bimodules. We will define a functor $$\mathbb{H}: D^{b}_ {\mathscr{B}}(\mathscr{X},\mathbb{Q})\rightarrow R\text{-}\mathrm{Mod}^{\mathbb{Z}}\text{-}R$$ given by $$\mathscr{F}\mapsto H_ {B}^{\ast}(\mathscr{X},\mathscr{F}):=\oplus_ {n\in\mathbb{Z}} H_ {\mathscr{B}}^{n}(\mathscr{X},\mathscr{F}).$$

Note that

  • $H_ {\mathscr{B}}^{\ast}(\text{pt},\mathbb{Q})\cong H_ {\mathscr{T}}^{\ast}(\text{pt},\mathbb{Q}) \cong R$.
  • $H_ {\mathscr{B}\times \mathscr{B}}^{\ast}(\text{pt},\mathbb{Q})\cong R\otimes_ {\mathbb{Q}}R$.
  • $H_ {\mathscr{B}}^{\ast}(\mathscr{X},\mathbb{Q})\cong H_ {\mathscr{B}\times\mathscr{B}}^{\ast}(\mathscr{G},\mathbb{Q})\leftarrow R\otimes_ {\mathbb{Q}}R$ is a graded homomorphism induced by $\mathscr{B}\backslash \mathscr{G}/\mathscr{B}\rightarrow \mathscr{B}\backslash\text{pt}/\mathscr{B}$.
  • By construction, $H_ {\mathscr{B}}^{\ast}(\mathscr{X},\mathbb{Q})$ acts on $H_ {\mathscr{B}}^{\ast}(\mathscr{F})$, therefore $\mathbb{H}(\mathscr{F})$ has an $R\otimes_ {\mathbb{Q}}R$-module structure.

Definition 1. For any $r\in \mathbb{Z}$, we define \begin{align} (r): R\text{-}\mathrm{Mod}^{\mathbb{Z}}\text{-}R &\rightarrow R\text{-}\mathrm{Mod}^{\mathbb{Z}}\text{-}R \\ M &\mapsto M(r),
\end{align} where $M(r)^{n}=M^{r+n}$.

So $\mathbb{H}\circ [1]\cong (1)\circ \mathbb{H}$.

Proposition 2. The functor $\mathbb{H}: \mathrm{IC}_ {\mathscr{B}}(\mathscr{X},\mathbb{Q})\hookrightarrow R\text{-}\mathrm{Mod}^{\mathbb{Z}}\text{-}R$ is fully faithful.

How to describe the essential image?

  • This is equivalent to describe $\mathbb{H}(\mathrm{IC}_ {w})$.
  • We describe a different family instead.

Definition 3. For any expression $\underline{w}:=(s_ {1}\dots s_ {r})$ into simple reflections, we set $$\mathrm{IC}_ {\underline{w}}:=\mathrm{IC}_ {s_ {1}}\ast\cdots \ast \mathrm{IC}_ {s_ {r}}.$$

Then

  1. $\mathrm{IC}_ {w}\subseteq\mathrm{IC}_ {\underline{w}} $ is a direct summand.
  2. $\mathrm{IC}_ {\mathscr{B}}(\mathscr{X},\mathbb{Q})$ is identified with full subcategory of $ D^{b}_ {\mathscr{B}}(\mathscr{X},\mathbb{Q})$ whose objects are direct sums of shifts of direct summands of $\mathrm{IC}_ {\underline{w}}$’s.

For any $s\in S$ simple reflection, consider the subalgebra $R^{s}\subseteq R$ of $s$-invariant elements, set $$\mathscr{B}_ {s}^{\text{bim}}:=R\otimes_ {R^{s}}R(1)\in R\text{-}\mathrm{Mod}^{\mathbb{Z}}\text{-}R.$$

Given any $\underline{w}:=(s_ {1}\dots s_ {r})$, we define $$\mathscr{B}_ {\underline{w}}^{\text{bim}}:=\mathscr{B}_ {s_ {1}}^{\text{bim}}\otimes_ {R}\cdots \otimes_ {R}\mathscr{B}_ {s_ {r}}^{\text{bim}}\cong R\otimes_ {R^{s_ {1}}}R\otimes_ {R^{s_ {2}}}\cdots\otimes_ {R^{s_ {r}}}R(r).$$

Proposition 4. For any $\underline{w}$, there is a canonical isomorphism $$\mathbb{H}(\mathrm{IC}_ {\underline{w}})\cong \mathscr{B}^{\text{bim}}_ {\underline{w}}.$$

Definition 5. The essential image of $\mathrm{IC}_ {\mathscr{B}}(\mathscr{X},\mathbb{Q})$ under $\mathbb{H}$ is the category of Soegel bimodules $\mathrm{SBim}(W,V)$ associated to

  • Coxeter system $(W,S)$,
  • representation $V:=\mathbb{Q}\otimes_ {\mathbb{Z}}\mathbb{X}_ {\ast}(T)$ of $W$.

General theory

Let $V$ be a finite dimensional representation of $W$ over some field $\mathbb{k}$. Define $\mathscr{T}\subseteq W$ the set of reflections (conjugates of elements in $S$).

Definition 6. We say $V$ is reflection faithful, if it is faithful and for any $w\in W$, we have that

$\text{dim}V^{w}=\text{dim}V-1$ if and only if $w\in \mathscr{T}$.

Example 7. Let $(W,S)$ be a Coxeter group. Set $V=\mathbb{R}^{S}$ with basis $(e_ {s}:s\in S)$. We define a symmetric bilinear form on $V$ by $$ \langle e_ {s},e_ {t} \rangle = \begin{cases} 1 & \text{if }s=t, \\ -1 & \text{if }s\neq t, \text{and }\langle s,t \rangle\text{infinite}, \\ \operatorname{cos}(\frac{\pi}{m_ {st}}) & \text{otherwise}. \end{cases} $$ Then the assignment $s\mapsto (\chi\mapsto \chi-2\langle \chi,e_ {s} \rangle)e_ {s}$ extends to a representation of $W$ on $V$. If $W$ is finite, then the pairing is non-degenerate, and this representation is reflection faithful.

Fix a Coxeter system $(W,S)$ and a finite dimensional representation $V$ of any field $\mathbb{k}$, such that $\text{char}(\mathbb{k})\neq 2$.

We set $R:=\mathrm{Sym}^{\ast}(V^{\ast})$ such that $V^{\ast}$ is in degree 2. We consider

  • $R\text{-}\mathrm{Mod}^{\mathbb{Z}}\text{-}R$,
  • $\mathscr{B}_ {s}^{\text{bim}}:=R\otimes_ {R^{s}}R(1)\in R\text{-}\mathrm{Mod}^{\mathbb{Z}}\text{-}R$
  • $\mathscr{B}_ {\underline{w}}^{\text{bim}}$ as before (Bott-Samelson modules).

Definition 8. Define $\mathrm{SB}^{\text{bim}}(W,V):=$ full subcategory of $R\text{-}\mathrm{Mod}^{\mathbb{Z}}\text{-}R$ whose objects are direct sums of grading shifts of direct summands of $\mathscr{B}_ {\underline{w}}^{\text{bim}}$ for $\underline{w}$ a word in $S$.

Remark 9. There is no reason why $\mathrm{SB}^{\text{bim}}(W,V)$ is well-behaved for general $V$. But when $V$ is reflection faithful, many properties which hold in geometric origin still hold.

Structure of $\mathrm{SB}^{\text{bim}}(W,V)$ when $V$ is reflection faithful

Theorem 10. When $V$ is reflection faithful, there exists a unique ring homomorphism $$\epsilon: \mathscr{H}_ {W,S}\rightarrow [\mathrm{SB}^{\text{bim}}(W,V)]_ {\oplus},$$ such that

  • $\epsilon(v)=[R(1)]$,
  • $\epsilon(\underline{H}_ {s}) = [\mathscr{B}_ {s}^{\text{bim}}]$ where $\underline{H}_ {s}$ is the Kazhdan-Lusztig basis element.
Proof.

For example, we check quadratic relations for $\mathscr{B}_ {s}^{\text{bim}}\otimes_ {R}\mathscr{B}_ {s}^{\text{bim}}\cong R\otimes_ {R^{s}}R\otimes_ {R^{s}}R(2).$

Lemma 11. Let $s$ act on $V$ by reflection and $\alpha\in V^{\ast}$ such that $s(\alpha)=-\alpha$, Then as graded $R^{s}$-algebra, $R$ is free with basis $\{1,\alpha\}$ with $\alpha$ in degree 2.

Granted this lemma, we see that $\mathscr{B}_ {s}^{\text{bim}}\cong R(1)\oplus R(-1)$ as $R^{s}$-modules. Therefore, $R\otimes_ {R^{s}}R\otimes_ {R^{s}}R(2)\cong \mathscr{B}_ {s}^{\text{bim}}(1)\oplus\mathscr{B}_ {s}^{\text{bim}}(-1).$ Therefore, $[\mathscr{B}_ {s}^{\text{bim}}]\cdot[\mathscr{B}_ {s}^{\text{bim}}]=(v+v^{-1})[\mathscr{B}_ {s}^{\text{bim}}].$

Proof of Lemma.

Write $V^{\ast} = H\oplus \mathbb{k}\alpha^{\ast}$, where $H$ is fixed by $s$. Then $R=\oplus_ {n\geq 0}\mathrm{Sym}^{\ast}(H)\alpha^{n}$ and hence, $$R^{s}=\oplus_ {n\geq 0,n\text{ even}}\mathrm{Sym}^{\ast}(H)\alpha^{n}.$$ This implies as graded left (or right) $R^{s}$-module we have that $R\cong R^{s}\oplus R^{s}(-2)$.

Similarly we can check for braid relations.

Theorem 12. For any $w\in W$, there exists a unique indecomposable object $\mathscr{B}_ {w}^{\text{bim}}\in \mathrm{SB}^{\text{bim}}(W,V)$ such that for any expression $\underline{w}$ of $w$, $\mathscr{B}_ {w}^{\text{bim}}$ is the unique indecomposable summand of $\mathscr{B}_ {\underline{w}}^{\text{bim}}$ which is not a summand of any $\mathscr{B}_ {\underline{y}}^{\text{bim}}(n)$ for any $\underline{y}$ expression of $y<w$, $n\in\mathbb{Z}$.

Further, \begin{align} W\times \mathbb{Z} & \rightarrow \mathrm{SB}^{\text{bim}}(W,V) \\ (w,n) &\mapsto \mathscr{B}_ {w}^{\text{bim}}(n) \end{align}

gives a bijection between $W\times \mathbb{Z}$ and isomorphism classes of indecomposable objects in $\mathscr{B}_ {w}^{\text{bim}}$.