Uniquenss of $M_ {w}$ and Failure of Lusztig's Conjecture

Our goal:

  • “Uniqueness” and properties of $M_ {w}$,
  • algebraic geometry:
    • exceptional collections,
    • attempts to categorify $f_ {w}$ and $M_ {w}$.

Counterexample to Lusztig’s original conjecture

Proposition 1. There exists examples of $M_ {w}$ which are not non-positive. For example,

  • in type $B_ {3}$ for $w=s_ {1}$,
  • in type $A_ {4}$ for $w=s_ {2}s_ {3}s_ {2}$. We have $M_ {s_ {2}s_ {3}s_ {2}}=V_ {0,p-1,p-1,0}-V_ {0,p-2,p-2,0}=L_ {0,p-1,p-1,0}-L_ {p-2,0,0,p-2}$.

Definition 2. Let $\gamma$ be a dominant weight. We say two weights $\lambda_ {1},\lambda_ {2}$ are $\gamma$-close, if $\lambda_ {1}-\lambda_ {2}$ is in the $W$-orbit of some dominant weight $\gamma’<\gamma$.

We say $\{M_ {w}’\}_ {w\in W}$ is a $\gamma$-close collection of candidate elements if

  1. for any irreducible unipotent representation of $G(\mathbb{F}_ {p})$, $\underline{\rho}=\sum_ {w\in \mathcal{J}}(\rho: R_ {\alpha_ {w}})M_ {w}’$.
  2. For any $w\in W$, $M_ {w}’$ is a linear combination of $V_ {\lambda}$ for $\lambda$ that is $\gamma$-close to $(p-1)\omega_ {I(M)}$.

Theorem 3. For $\gamma= k\rho$ where $k=2(\rho,\rho)+2$, our $\{M_ {w}\}$ is a $\gamma$-close collection of candidate elements.

Proposition 4. If $G=\operatorname{SL}(n)$ for $n\geq 3$, there exists some $\gamma$ such that for any $p$, the collection $\{M_ {w}\}$ is not the unique $\gamma$-close collection of candidate elements.

Proof for $G=\operatorname{SL}(3)$.

Write $A = V_ {1,1}+V_ {0,0}$. Note that $A= \mathrm{ch}(V_ {1,0}\otimes V_ {0,1})$ and $L_ {1,0}=V_ {1,0}$ and $L_ {0,1}=V_ {0,1}$. Bt the Steinberg tensor product formula, $A = L_ {p,1} = L_ {1,p}$ (for algebraic group representation $L_ {p,1}\cong L_ {0,1}\otimes \operatorname{Fr}^{\ast}L_ {1,0}$ but when restricted to $G(\mathbb{F}_ {p})$ finite group $\operatorname{Fr}^{\ast}$ is identity).

We can write $A=V_ {p,1}-V_ {p-2,2}+V_ {p-4,0} = V_ {1,p}-V_ {2,p-2}+V_ {0,p-4}$.

Now let $$ M_ {s_ {1}}^{\prime}:= M_ {s_ {1}}+A, $$ $$M_ {s_ {2}}^{\prime}:=M_ {s_ {2}}-A,$$ $$M_ {w}^{\prime}=M_ {w},\forall w\notin \{s_ {1},s_ {2}\}.$$

Let $\gamma = 4\rho.$

Lemma 5. Let $G=\operatorname{SL}(n)$for $1\leq n\leq 5$. If $c$ is some two-sided cell and $w,w’\in c$ are almost involutions, then $w=w’$ if and only if $I(w)=I(w’)$. (This is equivalent to say that any two standard Young tableaux of the same shape with identical descent sets must be equal if they have $\leq 5$ boxes.)

Proposition 6. Suppose that $G=\operatorname{SL}(n)$ for $n\leq 5$. For any fixed $\gamma$, there exists $p’>0$ such that if $p>p’$, and $\{M_ {w}^{\prime}\}_ {w\in W}$ is a $\gamma$-close collection of candidate elements, then for any $w\in W$, $M_ {w}^{\prime}-M_ {w}$ is a linear combination of $V_ {\lambda}$ for $\lambda$ which are $(\gamma+2\rho)$-close to $0$.

Proof.

Let $\gamma$ be a fixed dominant weight. Then we can choose $p’$ large so that if $p>p’$, then for any subset $I,I’\subseteq S$, there does not exist a weight $\lambda$ which is both $(\gamma+2\rho)$-close to $(p-1)\omega_ {I}$ and $(p-1)\omega_ {I^{\prime}}$. Since $\{M_ {w}\}_ {w\in W}$ and $\{M_ {w}^{\prime}\}_ {w\in W}$ satisfy the same equation, we have that $$\sum_ {w\in \mathcal{J}\cap c}(M_ {w}-M_ {w}^{\prime})=0.$$

For any $w\in W$, $M_ {w}^{\prime}-M_ {w}$ is a linear combination of $V_ {\lambda}$ with $\lambda$ being $\gamma$-close to $(p-1)\omega_ {I(w)}.$

Exercise: any such linear combination can be written as a linear combination of $L_ {\lambda}$ for $\lambda$ being $(\lambda+\rho)$-close to $(p-1)\omega_ {I(w)}$.

By Steinberg tensor product formula, this can be rewritten as a linear combination of $L_ {\lambda}$ for $\lambda$ being $p$-restricted and $(\gamma+2\rho)$-close to either $(p-1)\omega_ {I(w)}$ or to $0$.

We now claim that they are all close to $0$.

Suppose otherwise. Then some $L_ {\lambda}$ term for $\lambda$ being $(\gamma+\rho)$-close to $(p-1)\omega_ {I(w)}$ would also appear in $M_ {y}^{\prime}-M_ {y}$ for $y\neq w$ and $y\in c$. However, all $L_ {\lambda}$ term in $M_ {y}$ must be $(\gamma+2\rho)$-close to $(p-1)\omega_ {I(y)}$ or $0$. Since $q>q’$, we conclude that $\omega_ {I(y)}=\omega_ {I(w)}$, then $y=w$ by the lemma above.

Definition 7. Consider a family of collections $\{M_ {w}\}_ {w\in W}$ defined simultaneously across all large $p$ (we write $M_ {w}^{p}$). We say $M_ {w}$ is a linear combination of Weyl characters depending on $p$, if there exist polynomials $f_ {i}(t)\in \mathbb{Z}[t][\mathbb{X}^{\ast}(T)]$ such that $M_ {w}^{p}=\sum_ {i} c_ {i}V_ {f_ {i}(p)}$ for all $p$ large.

Corollary 8. For $G=\operatorname{SL}(5)$ and any dominant weight $\gamma$, there exists no $\gamma$-close collection of candidate elements $\{M_ {w}^{\prime}\}_ {w\in W}$ with each $M_ {w}^{\prime}$ a linear combination of Weyl characters depending on $p$ which satisfies Lusztig’s properties.

Proof.

The failure of positivity in type $A_ {4}$ can not be resolved by any linear combination of $V_ {\lambda}$ for $\lambda$ close to $0$ if $p>\!\!>0$. So there is no $\{M_ {w}^{\prime}\}$ satisfying all assumptions of Lusztig conjecture, since it would differ from $\{M_ {w}\}$ by such linear combination.

Geometry

Let $\mathbb{k}=\overline{\mathbb{F}_ {p}}$ and $q=p^{r}$ a $p$-power.

Lemma 9. Let $X$ be a projective variety over $\mathbb{F}_ {q}$ with an action of an algebraic group $G$. Then we can consider the representation $\mathbb{k}[X(\mathbb{F}_ {q})]$ of $G(\mathbb{F}_ {q})$. Suppose that we have a decomposition $$[\mathscr{O}_ {\Delta}] = \sum_ {i}[\mathscr{F}_ {i}\boxtimes \mathscr{F}_ {i}’]$$ in $K_ {0}(\operatorname{Coh}^{G}(X\times X))$. Then let the virtual $\mathbb{k}$-representation of $G$ given by $$V= \sum_ {i} R\Gamma(\mathscr{F}_ {i}\otimes \operatorname{Fr}^{\ast}\mathscr{F}_ {i}^{\prime}).$$ Then $V|_ {G(\mathbb{F}_ {q})} = \mathbb{k}[X(\mathbb{F}_ {q})]$.

Proof.

Let $\Delta: X\rightarrow X\times X$ be the diagonal morphism. Let $g: X\rightarrow X\times X$ be the graph of Frobenius $x\mapsto (x,\operatorname{Fr}(x))$. Then $\Delta$ is equivariant for the diagonal action of $G$ on $X\times X$ and $g$ is equivariant is equivariant for $G$ by standard action on first factor and standard action twisted by Frob on the second factor. Therefore, both morphisms are equivariant for diagonal action of $G(\mathbb{F}_ {q})$.

Apply $g^{\ast}$ to objects in the two sides of decomposition of $\mathscr{O}_ {\Delta}$. On right-hand-side, one get $$g^{\ast}(\mathscr{F}_ {i}\boxtimes \mathscr{F}_ {i}^{\prime})\cong \mathscr{F}_ {i}\otimes \operatorname{Fr}^{\ast}(\mathscr{F}_ {i}^{\prime})$$ and on left-hand-side $$g^{\ast}\mathscr{O}_ {\Delta}\cong \mathscr{O}_ {\Delta(X)\cap g(X)}$$ where $\Delta(X)\cap g(X)$ is a disjoint union of points in $X(\mathbb{F}_ {q})$. Therefore, $$[\mathscr{O}_ {X(\mathbb{F}_ {q})}] = \sum_ {i}[\mathscr{F}_ {i}\otimes \operatorname{Fr}^{\ast}(\mathscr{F}_ {i}^{\prime})]$$ in the $G(\mathbb{F}_ {q})$-equivariant Grothendieck group of $X$. Now apple $\Gamma$, one gets the desired result.

Some Category Theory

Let $D$ be a triangulated category over a field $\mathbb{k}$.

Definition 10. We say $\mathscr{F}\in D$ is exceptional, if there is an isomorphism of graded $\mathbb{k}$-algebras $\mathrm{Hom}_ {D}^{\bullet}(F,F)\cong \mathbb{k}$. We say a collection $(\mathscr{F}_ {1},\dots, \mathscr{F}_ {n})$ of exceptional objects is exceptional, if $1\leq i<j\leq n$, one has $$\mathrm{Hom}_ {D}^{\bullet}(\mathscr{F}_ {i},\mathscr{F}_ {j})=0.$$ It is a full exceptional collection if furthermore $\mathscr{F}_ {i}$ generate $D$ as a triangulated category.

Definition 11. A collection $(\mathscr{G}_ {1},\dots,\mathscr{G}_ {n})$ is the dual exceptional collection to $(\mathscr{F}_ {1},\dots, \mathscr{F}_ {n})$ if

\begin{equation} \mathrm{RHom}(\mathscr{F}_ {i},G_ {j}[l])=\mathrm{RHom}(\mathscr{G}_ {i}, \mathscr{F}_ {j}[l])= \begin{cases} \mathbb{k} & \text{if } i=j, l=0, \\ 0 & \text{otherwise}. \end{cases} \end{equation}

By definition, the dual of an exceptional collection is also an exceptional collection.