VIII. Hall algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 13 October 2012

The results in §1 are outlines in [CPr1994] §9.3D. The proof of Theorem (1.2) appears in [BGP1973] Theorem 3.1 The results in §1 are outlined in [CPr1994] §9.3D. The proof of Theorem (1.2) appears in [BGP1973] Theorem 3.1 and the proof of Theorem (1.4) appears in [Lus1990] Prop. 5.7. The material in §2 is a combination of [Lus1991] and [Lust1993] Part II. In particular, Theorem (2.7)(1) is proved in [Lus1993] 13.1.2, 12.3.2, and 9.2.7, Theorem (2.7)(2) is proved in [Lus1993] 13.1.5, 13.1.12e, 12.3.3, and 9.2.11, Theorem (2.7)(3) is proved in [Lus1993] 13.1.12d and 12.3.6. The statement about the symmetric form given in (2.6) is proved in [Lus1993] 12.2.2, 9.2.9 and the references given there. The proof of the isomorphism theorem in (2.8) is given in [Lus1993] 13.2.11 and in [Lus1991] Th. 10.17. The material in §3 appears in [Lus1990] §9. The isomorphism in (3.4) is stated in [Lus1990] 9.6.

Hall algebras

The Hall algebra is an algebra which has a basis labelled by representations of quivers and for which the structure constants with respect to this basis reflect the structure of these representations. The Hall algebra encodes a large amount of information about the representations of the quiver. Amazingly, this algebra is almost isomorphic to the nonnegative part of the quantum group.

Quivers

A quiver is an oriented graph $\Gamma ,$ i.e. a set of vertices and directed edges. The following is an example of a quiver.

$$$$

Every Dynkin diagram if type $A,$ $D$ or $E$ can be made into a quiver by orienting the edges. Note that there are many possible ways of orienting the edges of a Dynkin diagram in order to make a quiver. For example the quivers

$$\begin{array}{cc} & \end{array}$$

are both obtained by orienting the edges of the Dynkin diagram of type $E_6\text{.}$

Representations of a quiver

A representation $R$ of a quiver $\Gamma$ over a field $k$ is a labelling of the graph $\Gamma$ such that

1. Each vertex $i\in \Gamma$ is labelled by a vector space $R_i$ over $k,$
2. Each edge $i\to j$ in $\Gamma$ is labelled by a (vector space) homomorphism ${\varphi }_{ij}:R_i\to R_j\text{.}$

Define morphisms of representations of quivers in the natural way and make the category of representations of the quiver $\Gamma \text{.}$ The dimension of a representation $R$ is the vector $\text{dim}\left(R\right)=\left(d_i\right)$ where, for each vertex $i\in \Gamma ,$ $d_i=\text{dim}\left(R_i\right)\text{.}$ An irreducible representation of $\Gamma$ is a representation $R$ of $\Gamma$ such that the only subrepresentations of $R$ are 0 and $R\text{.}$

A representation of $R$ of a quiver $\Gamma$ is indecomposable if it cannot be written as $R=S\oplus T$ where $S$ and $T$ are nonzero representations of $\Gamma \text{.}$

Let $\Gamma$ be a quiver.

1. There are a finite number of indecomposable representations of $\Gamma$ if and only if $\Gamma$ is an oriented Dynkin diagram of type $A,$ $D$ or $E\text{.}$
2. If $\Gamma$ is an oriented Dynkin diagram of type $A,$ $D$ or $E$ then the indecomposable representations of $\Gamma$ are in 1-1 correspondence with the positive roots for the Lie algebra $𝔤$ corresponding to the Dynkin diagram.

Definition of the Hall algebra

Let $\Gamma$ be a quiver and let ${𝔽}_q$ be a finite field with $q$ elements. The Hall algebra or Grothendieck ring $R\Gamma$ of representations of $\Gamma$ is the algebra over $ℂ$ with

1. basis labelled by the isomorphism classes $\left[R\right]$ of representations of $\Gamma$ over ${𝔽}_q,$ and
2. multiplication of two isomorphism classes $\left[R\right]$ and $\left[S\right]$ given by $$\left[R\right]·\left[S\right]=\sum _{\left[T\right]}{C}_{RS}^T\left[T\right]\text{where}{C}_{RS}^T=\text{Card}\left(\{P\subseteq T\mid P\cong R,T/P\cong S\}\right)\text{.}$$

Connecting Hall algebras to the quantum group

Let $\Gamma$ be a quiver which is obtained by orienting the edges of a Dynkin diagram of type $A,$ $D,$ or $E,$ and let ${𝔽}_q$ be a finite field with $q$ elements. Let us describe explicitly two types of indecomposable representations of $\Gamma \text{.}$

1. Let $i$ be a vertex of $\Gamma \text{.}$ The representation $$e_i\text{given by}{V}_j=\{\begin{array}{cc}{𝔽}_q,& \text{if}j=i;\\ 0,& \text{if}j\ne i;\end{array}$$ is an irreducible representation of $\Gamma \text{.}$
2. Let $i\to j$ be an edge of $\Gamma \text{.}$ The representation $$e_{ij}\text{given by}{V}_{\ell }=\{\begin{array}{cc}{𝔽}_q,& \text{if}\ell =i\text{or}\ell =j;\\ 0,& \text{otherwise;}\end{array}\text{and}{\varphi }_{ij}={\text{id}}_{{𝔽}_q},$$ is an indecomposable (but not irreducible) representation of $\Gamma \text{.}$

The following relations hold in the Hall algebra $R\Gamma ,$

$$e_{ij}=e_i{e}_j-e_j{e}_i,e_i{e}_{ij}=q{e}_{ij}{e}_i,e_{ij}{e}_i=q{e}_j{e}_{ij},\text{for each edge}i\to j\text{in}\Gamma \text{.}$$

It is easier to prove the first relation by writing it in the form $e_i{e}_j=e_{ij}+e_j{e}_i\text{.}$ Combining the first two of these relations and the first and last of these relations respectively, gives the identities

$$e_i^2{e}_j-(q+1)e_i{e}_j{e}_i+q{e}_j{e}_i^2=0\text{and}{e}_i{e}_j^2-(q+1)e_j{e}_i{e}_j+q{e}_j^2{e}_i=0,\text{respectively.}$$

We shall make the Hall algebra a bit bigger by adding the $K_i^{\pm 1}\text{s}$ that are in the quantum group $U_q𝔤\text{.}$ Let $𝔤$ be the finite dimensional complex simple Lie algebra corresponding to the Dynkin diagram given by $\Gamma$ and let $U_q𝔤$ be the rational version of the quantum group with $k=ℂ$ and $q\in ℂ$ the number of elements in the field ${𝔽}_q\text{.}$ Let $U_q𝔥$ be the subalgebra of $U_q𝔤$ generated by $K_1^{\pm 1},\dots ,K_r^{\pm 1}\text{.}$ Let ${\alpha }_1,\dots ,{\alpha }_r$ be the simple roots corresponding to the Lie algebra $𝔤$ (see II (2.6)). Define

$$\begin{array}{c}\widetilde{R\Gamma }=\text{algebra generated by}R\Gamma \text{and}{K}_1^{\pm 1},\dots ,K_r^{\pm 1}\text{with the additional relations}\\ K_i\left[R\right]K_i^{-1}=q^{({\alpha }_i,d\left(R\right))}\left[R\right],\text{for all}1\le i\le r\text{and representations}R\text{of}\Gamma ,\end{array}$$

where $d\left(R\right)=\sum _{j=1}^r\text{dim}\left(R_j\right){\alpha }_j,$ and the inner product in the exponent of $q$ is the inner product on ${𝔥}_{ℝ}^{*}$ given in II (2.7).

Let $\Gamma$ be a quiver which is obtained by orienting the edges of a Dynkin diagram of type $A,$ $D$ or $E\text{.}$ Let $R\Gamma$ be the Hall algebra of representations of $\Gamma$ over the finite field ${𝔽}_q$ with $q$ elements and let $\widetilde{R\Gamma }$ be the extended Hall algebra defined above. Let $U_q𝔤$ be the rational form of the quantum group with $k=ℂ$ which corresponds to the Dynkin diagram $\Gamma$ and let

$$U_q{𝔟}^{+}=\text{subalgebra of}{U}_q𝔤\text{generated by}{K}_1^{\pm 1},\dots ,K_r^{\pm 1}\text{and}{E}_1,\dots ,E_r\text{.}$$

Choose elements $z_1,\dots ,z_r\in \mathbb{Z}$ such that $z_i-z_j=1$ if $i\to j$ is an edge in $\Gamma \text{.}$ Then the homomorphism of algebras determined by

$$\begin{array}{ccc}{U}_q{𝔟}^{+}& ⟶& \widetilde{R\Gamma }\\ \phantom{A}\\ K_i^{\pm 1}& ⟼& K_i^{\pm 1}\\ E_i& ⟼& K_i^{z_i}{e}_i\end{array}$$

is an isomorphism.

An algebra of perverse sheaves

In this section we shall construct an algebra $𝒦$ from a Dynkin diagram $\Gamma \text{.}$ There is a strong relationship between this algebra and the quantum group $U_q𝔤$ where $𝔤$ is the simple complex Lie algebra corresponding to the Dynkin diagram $\Gamma \text{.}$

The algebra $𝒦$ is graded,

$$𝒦=\underset{\nu \in Q^{+}}{⨁}{𝒦}_{\nu },$$

in the same way that the quantum group $U_q{𝔫}^{+}$ is graded, see VII (1.2). The vector space $𝒦$ comes with natural shift maps $\left[n\right]$ which correspond to multiplication by $q^n$ in the quantum group $U_q{𝔟}^{+}\text{.}$ The algebra $𝒦$ has a natural multiplication which comes from an induction functor and a natural "psuedo-comultiplication" which comes from a restriction functor. The multiplication and the pseudo-comultiplication turn out to be almost the same as the multiplication and the comultiplication on the quantum group $U_q{𝔟}^{+}\text{.}$ Lastly, the algebra $𝒦$ has a natural inner product $\{,\}$ that is related to the inner product $⟨,⟩$ pairing $U_q{𝔟}^{-}$ and $U_q{𝔟}^{+},$ (see VII (2.1)).

In Theorem (2.8) we shall see that if we extend the algebra $𝒦$ a little bit, by adding the $K_i^{\pm 1}\text{'s}$ that are in the quantum group $U_q𝔤$ then we get an algebra $\tilde{K}$ such that

$$\widetilde{𝒦}\simeq U_q{𝔟}^{+}\text{.}$$

This last fact is very similar to the case of the Hall algebra (1.4) where after extending the Hall algebra $R\Gamma$ by adding the $K_i^{\pm 1}\text{'s}$ that are in the quantum group $U_q𝔤,$ we got an algebra $\widetilde{R\Gamma }$ which was also isomorphic to $U_q{𝔟}^{+}\text{.}$ We shall see in section 3 that this is not a coincidence, there is a concrete connection between $R\Gamma$ and the algebra $𝒦\text{.}$ The advantage of working with the algebra $𝒦$ instead of the Hall algebra $R\Gamma$ is that $𝒦$ has more natural structure than $R\Gamma ,$ it has:

1. a natural pseudo-comultiplication $r:𝒦\to 𝒦\otimes 𝒦,$
2. a natural inner product $\{,\}:𝒦\times 𝒦\to \mathbb{Z}\left(\left(q\right)\right),$
3. a natural involution $D:𝒦\to 𝒦,$
4. a natural basis coming from simple perverse sheaves.

The natural basis coming from simple perverse sheaves is called the canonical basis.

$\Gamma \text{-graded}$ vector spaces and the varieties $E_V$ with $G_V$ action

Let $\Gamma$ be a quiver obtained by orienting the edges of a Dynkin diagram of type $A,$ $D$ or $E\text{.}$ For convenience we label the vertices by $1,2,\dots ,r\text{.}$ Let $𝔤$ be the finite dimensional complex simple Lie algebra corresponding to the Dynkin diagram given by $\Gamma \text{.}$

Let $p$ be a positive prime integer and let $\overline{{𝔽}_p}$ be the algebraic closure of the finite field ${𝔽}_p$ with $p$ elements. A $\Gamma \text{-graded}$ vector space $V$ over $\overline{{𝔽}_p}$ is a labelling of the graph $\Gamma$ such that each vertex $i$ is labelled by a vector space $V_i$ over $\overline{{𝔽}_p}\text{.}$ The dimension of a $\Gamma \text{-graded}$ vector space $V$ is the $r\text{-tuple}$ of nonnegative integers $\text{dim}\left(V\right)=\left(\text{dim}\left(V_i\right)\right)\text{.}$ We shall identify dimensions of $\Gamma \text{-graded}$ vector spaces with elements of

$$Q^{+}=\sum _i\mathbb{N}{\alpha }_i\text{so that}\text{dim}\left(V\right)=\sum _{i=1}^r\text{dim}\left(V_i\right){\alpha }_i,$$

where ${\alpha }_1,\dots ,{\alpha }_r$ are the simple roots for $𝔤$ and $\mathbb{N}={\mathbb{Z}}_{\ge 0}\text{.}$

Fix an element $\nu \in Q^{+}$ and a $\Gamma \text{-graded}$ vector space $V$ over $\overline{{𝔽}_p}$ such that $\text{dim}\left(V\right)=\nu \text{.}$ Define

$$G_V=\prod _{i}GL\left(V_i\right)\text{and}{E}_V=\underset{i\to j}{⨁}\text{Hom}(V_i,V_j),$$

where the sum in the definition of $E_V$ is over all edges of $\Gamma \text{.}$ There is a natural action of $G_V$ on $E_V$ given by

$$g·\left({\varphi }_{ij}\right)=\left(g_j{\varphi }_{ij}{g}_i^{-1}\right),\text{if}\left({\varphi }_{ij}\right)\in E_V\text{and}g=(g_1,\dots ,g_r)\in G_V\text{.}$$

Let $x\in E_V$ and let $W$ be a $\Gamma \text{-graded}$ subspace of $V,$ i.e. $W_i\subseteq V_i$ for all vertices $i$ in $\Gamma \text{.}$ The subspace $W$ is $x\text{-stable}$ if $x{W}_i\subseteq W_j$ for all edges $i\to j$ in $\Gamma \text{.}$ We shall simply write $W\subseteq V$ if $W$ is a $\Gamma \text{-graded}$ subspace of $V$ and $xW\subseteq W$ if $W$ is $x\text{-stable.}$

Definition of the categories ${𝒬}_V$ and ${𝒬}_T\otimes {𝒬}_W$

The reader map ship this definition if it looks like too much to swallow. The only important thing at this stage is that ${𝒬}_V$ is a category of objects and it is contained in a category called $D_c^b\left(E_V\right)\text{.}$

Let $V$ be a $\Gamma \text{-graded}$ vector space over $\overline{{𝔽}_p}$ and let $E_V$ be the variety over $\overline{{𝔽}_p}$ defined in (2.1). Let $D_c^b\left(E_V\right)$ be the bounded derived category of ${\bar{\mathbb{Q}}}_l\text{-(constructible)}$ sheaves on $E_V,$ see IV (1.4). Recall that $D_c^b\left(E_V\right)$ comes endowed with shift functors IV (2.4),

$$\begin{array}{cccc}\left[n\right]:& D_c^b\left(E_V\right)& ⟶& D_c^b\left(E_V\right)\\ & A& ⟼& A\left[n\right]\text{.}\end{array}$$

Define

$$\begin{array}{ccc}{𝒬}_V& =& \text{the full subcategory of}{D}_c^b\left(E_V\right)\text{consisting of finite direct sums of simple}\\ & & \text{perverse sheaves}L\text{such that some shift of}L\text{is a direct summand of}{L}_{\overrightarrow{\nu }}\\ & & \text{for some partition}\overrightarrow{\nu }\text{of}\nu =\text{dim}\left(V\right)\text{.}\end{array}$$

The complexes $L_{\overrightarrow{\nu }}$ are defined in (2.7). Let $T$ and $W$ be a $\Gamma \text{-graded}$ vector spaces of $\overline{{𝔽}_p}\text{.}$ Define

$$\begin{array}{ccc}{𝒬}_T\otimes {𝒬}_W& =& \text{the complexes}L\in D_c^b(E_T\times E_W)\text{such that}L\cong \underset{i=1}{\overset{s}{⨁}}{A}_i\otimes B_i,\\ & & \text{for some}{A}_i\in {𝒬}_T,B_i\in {𝒬}_W,\text{and some positive integer}s\text{.}\end{array}$$

This is a subcategory of $D_c^b(E_T\times E_W)\text{.}$

The Grothendieck group $𝒦$ associated to the categories ${𝒬}_V$

Let $\nu \in Q^{+}$ and let $V$ be a $\Gamma \text{-graded}$ vector space of dimension $\nu \text{.}$ Let ${𝒬}_V$ be as in (2.2). The important thing about ${𝒬}_V$ at the moment is that it is a category related to $E_V\text{.}$

The Grothendieck group $𝒦\left({𝒬}_V\right)$ of the category ${𝒬}_V$ is the $ℂ\left(q\right)\text{-module}$ generated by the isomorphism classes of objects in ${𝒬}_V$ with the addition operation given by the relations

$$[B_1\oplus B_2]=\left[B_1\right]+\left[B_2\right],\text{if}{B}_1,B_2\in {𝒬}_V,$$

and multiplication by $q$ given by the relations

$$\left[B\left[n\right]\right]=q^n\left[B\right],\text{for}B\in {𝒬}_V\text{and}n\in \mathbb{Z},$$

where the map $B\to B\left[n\right]$ is the shift functor on $D_c^b\left(E_V\right),$ see IV (2.4). The structure of $𝒦\left({𝒬}_V\right)$ depends only on the element $\nu$ and so we shall often write ${𝒦}_{\nu }$ in place of $𝒦\left({𝒬}_V\right)\text{.}$

Define

$$𝒦=\underset{\nu \in Q^{+}}{⨁}{𝒦}_{\nu }\text{.}$$

The group $𝒦$ is graded in the same way that ${𝔘}_q{𝔫}^{+}$ is graded, see VII (1.2).

Definition of the multiplication in $𝒦$

Let $V$ be a $\Gamma \text{-graded}$ vector space. Let $T$ and $W$ be $\Gamma \text{-graded}$ vector spaces such that

$$W\subseteq V\text{and}V/W\cong T\text{.}$$

If $x\in E_V$ such that $xW\subseteq W$ then let $x_W$ be the linear transformation of $W$ induced by the action of $x$ on $W$ and let $x_T$ be the linear transformation of $T\cong V/W$ induced by the action of $x$ on $V/W\text{.}$ Define

$$\begin{array}{c}𝒮=\{x\in E_V\mid xW\subseteq W\},\\ P=\{g\in G_V\mid gW\subseteq W\},U=\{g\in P\mid g_W={\text{id}}_W,g_T={\text{id}}_T\}\text{.}\end{array}$$

The groups $P$ and $U$ are subgroups of $G_V\text{.}$ The group $P$ is the stabilizer of $W$ in $G_V,$ it is a parabolic subgroup of $G_V\text{.}$ The group $U$ is the unipotent radical of $P\text{.}$

Let ${𝒬}_T\otimes {𝒬}_W$ be the subcategory of $D_c^b(E_T\otimes E_W)$ which is defined in (2.2). The diagram

$$\begin{array}{ccccccc}{E}_T\times E_W& \stackrel{p_1}{⟵}& G_V{\times }_U𝒮& \stackrel{p_2}{⟶}& G_V{\times }_P𝒮& \stackrel{p_3}{⟶}& E_V\\ (x_T,x_W)& ⟼ & (g,x)& ⟼& (g,x)& ⟼& gx\end{array}$$

induces the diagram

$$\begin{array}{ccccccccc}{𝒬}_T\otimes {𝒬}_W& ⟶& D_c^b(E_T\times E_W)& \stackrel{p_1^{*}}{⟶}& D_c^b\left(G{\times }_U𝒮\right)& \stackrel{{\left(p_2\right)}_{♭}}{⟶}& D_c^b\left(G{\times }_P𝒮\right)& \stackrel{{\left(p_3\right)}_{!}}{⟶}& D_c^b\left(E_V\right)\end{array}$$

where the first map is the inclusion map.

Let $V$ be a $\Gamma \text{-graded}$ vector space and let $E_V$ be the variety with the $G_V$ action which is defined in (2.1). Let $W$ and $T$ be $\Gamma \text{-graded}$ vector spaces such that $W\subseteq V$ and $V/W\cong T\text{.}$ Let ${𝒬}_T\otimes {𝒬}_W$ and ${𝒬}_V$ be the categories of complexes of sheaves on $E_T\times E_W$ and $E_V,$ respectively, which are defined in (2.2). There is a well defined functor

$$\begin{array}{cccc}{\text{Ind}}_{T,W}^V:& {𝒬}_T\otimes {𝒬}_W& ⟶& {𝒬}_V\\ & A& ⟼& \left({\left(p_3\right)}_{!}{\left(p_2\right)}_{♭}{p}_1^{*}A\right)[\text{dim}\left(p_1\right)-\text{dim}\left(p_2\right)]\end{array}$$

where $p_1,$ $p_2,$ and $p_3$ are as defined in the diagram above, $\text{dim}\left(p_1\right)$ is the dimension of the fibers of the map $p_1,$ and $\text{dim}\left(p_2\right)$ is the dimension of the fibers of the map $p_2\text{.}$

The multiplication in $𝒦$ is defined by the formula

$$\left[A\right]·\left[B\right]=\left[{\text{Ind}}_{T,W}^V(A\otimes B)\right],\text{for}A\in {𝒬}_T\text{and}B\in {𝒬}_W\text{.}$$

With this multiplication $𝒦$ becomes an algebra. The strange shift by $[\text{dim}\left(p_1\right)-\text{dim}\left(p_2\right)]$ in the definition of ${\text{Ind}}_{T,W}^V$ is there to make the multiplication in $𝒦$ match up with the multiplication in the nonnegative part of the quantum group $U_q{𝔟}^{+},$ see Theorem (2.8) below.

Definition of the psuedo-comultiplication $r:𝒦\to 𝒦\otimes 𝒦$

Let $V$ be a $\Gamma \text{-graded}$ vector space. Let $T$ and $W$ be $\Gamma \text{-graded}$ vector spaces such that

$$W\subseteq V\text{and}V/W\cong T\text{.}$$

If $x\in E_V$ such that $xW\subseteq W$ then let $x_W$ be the linear transformation of $W$ induced by the action of $x$ on $W$ and let $x_T$ be the linear transformation of $T\cong V/W$ induced by the action of $x$ on $V/W\text{.}$

Define

$$𝒮=\{x\in E_V\mid xW\subseteq W\}$$

and let ${𝒬}_V$ be the subcategory of $D_c^b\left(E_V\right)$ which is defined in (2.2). The diagram

$$\begin{array}{ccccc}{E}_V& \stackrel{\iota }{⟵}& 𝒮& \stackrel{\kappa }{⟶}& E_T\times E_W\\ x& ⟼ & x& ⟼& (x_T,x_W)\end{array}$$

induces the diagram

$$\begin{array}{ccccccc}{𝒬}_V& ⟶& D_c^b\left(E_V\right)& \stackrel{{\iota }^{*}}{⟶}& D_c^b\left(𝒮\right)& \stackrel{{\kappa }_{!}}{⟶}& D_c^b(E_T\times E_W)\end{array}$$

where the first map is the inclusion map.

Let $V$ be a $\Gamma \text{-graded}$ vector space and let $E_V$ be the variety with the $G_V$ action which is defined in (2.1). Let $W$ and $T$ be $\Gamma \text{-graded}$ vector spaces such that $W\subseteq V$ and $V/W\cong T\text{.}$ Let ${𝒬}_T\otimes {𝒬}_W$ and ${𝒬}_V$ be the categories of complexes of sheaves on $E_T\times E_W$ and $E_V,$ respectively, which are defined in (2.2). There is a well defined functor

$$\begin{array}{cccc}{\text{Res}}_{T,W}^V:& {𝒬}_V& ⟶& {𝒬}_T\otimes {𝒬}_W\\ & B& ⟼& \left({\kappa }_{!}{\iota }^{*}B\right)[\text{dim}\left(p_1\right)-\text{dim}\left(p_2\right)-2\text{dim}(G_V/P)]\end{array}$$

where $p_1,$ $p_2,$ $\kappa ,$ and $\iota$ are as defined above, $\text{dim}\left(p_1\right)$ is the dimension of the fibers of the map $p_1,$ $\text{dim}\left(p_2\right)$ is the dimension of the fibers of the map $p_2,$ and $P$ is the parabolic subgroup of $G_V$ defined in (2.4).

The pesudo-comultiplication on $𝒦$ is the map $r:𝒦\to 𝒦\otimes 𝒦$ defined by

$$r\left(\left[A\right]\right)=\left[{\text{Res}}_{T,W}^V\left(A\right)\right],\text{if}A\in {𝒬}_V\text{.}$$

The strange shift by $[\text{dim}\left(p_1\right)-\text{dim}\left(p_2\right)-2\text{dim}(G_V/P)]$ in the definition of ${\text{Res}}_{T,W}^V$ is there to make the pseudo-comultiplication in $𝒦$ match up with the comultiplication in the nonnegative part of the quantum group $U_q{𝔟}^{+},$ see Theorem (2.8) below.

The symmetric form on $𝒦$

Recall that we write ${𝒦}_{\nu }$ in place of $𝒦\left({𝒬}_V\right)$ since the structure of $𝒦\left({𝒬}_V\right)$ depends only on $\nu \text{.}$ For each $\nu \in Q^{+},$ define a bilinear form

$$\begin{array}{c}{\{,\}}_{\nu }:{𝒦}_{\nu }\times {𝒦}_{\nu }\to ℂ\left(q\right)\text{by defining}\\ {\{\left[B_1\right],\left[B_2\right]\}}_{\nu }=\sum _j{q}^{-j}\text{dim}\left({\mathscr{H}}^{j+2\text{dim}(G\backslash \Omega )}\left(u_{!}(t_{♭}{s}^{*}{B}_1\otimes t_{♭}{s}^{*}{B}_2)\right)\right),\end{array}$$

for $B_1,B_2\in {𝒬}_V\text{.}$ The vector spaces ${\mathscr{H}}^{j+2\text{dim}(G\backslash \Omega )}\left(u_{!}(t_{♭}{s}^{*}{B}_1\otimes t_{♭}{s}^{*}{B}_2)\right)$ are defined in (2.10) below. At this stage the important thing is that they depend only on $B_1,$ $B_2$ and $j\text{.}$

Use the forms ${\{,\}}_{\nu },$ $\nu \in Q^{+},$ to define a bilinear form

$$\begin{array}{c}\{,\}:𝒦\times 𝒦\to \mathbb{Z}\left(\left(q\right)\right)\text{on}𝒦=\underset{\nu \in Q^{+}}{⨁}{𝒦}_{\nu }\text{by setting}\\ \phantom{A}\\ \begin{array}{cccc}\{{𝒦}_{\mu },{𝒦}_{\nu }\}& =& 0,& \text{if}\mu ,\nu \in Q^{+}\text{such that}\mu \ne \nu ,\text{and}\\ \{x,y\}& =& {\{x,y\}}_{\nu },& \text{if}x,y\in {𝒦}_{\nu }\text{.}\end{array}\end{array}$$

Let $V$ be a $\Gamma \text{-graded}$ vector space and let $T$ and $W$ be $\Gamma \text{-graded}$ subspaces such that $W\subseteq V$ and $T\cong V/W\text{.}$ Let $A\in {𝒬}_T\otimes {𝒬}_W$ and let $B\in {𝒬}_V\text{.}$ Then

$$\begin{array}{ccc}\{A,{\text{Res}}_{T,W}^V\left(B\right)\}& =& \{{\text{Ind}}_{T,W}^V\left(A\right),B\}\end{array}$$

The result in this theorem is an analogue of the property of the bilinear form $⟨,⟩$ on the quantum group which is given in VII (2.1)(d).

Definition of the elements $L_{\overrightarrow{\nu }}\in 𝒦$

Let $\nu \in Q^{+}$ and let $V$ be a $\Gamma \text{-graded}$ subspace of dimension $\nu \text{.}$ A partition of $\nu$ is a sequence $\overrightarrow{\nu }=({\nu }^1,\dots ,{\nu }^m)$ of elements of the root lattice $Q$ such that

1. each ${\nu }^j,$ $1\le j\le m,$ is a nonnegative integer multiple of a simple root, and
2. ${\nu }^1+\dots +{\nu }^m=\nu \text{.}$

For example we might have $\overrightarrow{\nu }=(3{\alpha }_1,2{\alpha }_3,0,{\alpha }_1,2{\alpha }_1)$ if $\nu =6{\alpha }_1+2{\alpha }_3\text{.}$ A flag of type $\overrightarrow{\nu }$ in $V$ is a sequence

$$f=(V=V^{\left(0\right)}\supseteq V^{\left(1\right)}\supseteq \dots \supseteq V^{\left(m\right)}=0)$$

of $\Gamma \text{-graded}$ subspaces of $V$ such that $\text{dim}(V^{(\ell -1)}/V^{\left(\ell \right)})={\nu }^{\ell },$ for all $1\le \ell \le m\text{.}$

Let $x\in E_V\text{.}$ A flag $f$ is $x\text{-stable}$ if $x{V}^{\left(\ell \right)}\subseteq V^{\left(\ell \right)}$ for all $1\le \ell \le m\text{.}$ Define

$${ℱ}_{\overrightarrow{\nu }}=\{(x,f)\mid x\in E_V,f\text{is an}x\text{-stable flag of type}\overrightarrow{\nu }\text{in}V\}\text{.}$$

The map

$$\begin{array}{ccc}\begin{array}{ccc}{ℱ}_{\overrightarrow{\nu }}& \stackrel{{\pi }_{\overrightarrow{\nu }}}{\to }& E_V\\ (x,f)& ⟼& x\end{array}& \text{induces a map}& \begin{array}{ccc}{D}_c^b\left({ℱ}_{\overrightarrow{\nu }}\right)& \stackrel{{\left({\pi }_{\overrightarrow{\nu }}\right)}_{!}}{⟶}& D_c^b\left(E_V\right)\text{.}\end{array}\end{array}$$

Let $f\left(\overrightarrow{\nu }\right)=\text{dim}\left({ℱ}_{\overrightarrow{\nu }}\right)$ and define

$$\begin{array}{c}{L}_{\overrightarrow{\nu }}=\left({\left({\pi }_{\overrightarrow{\nu }}\right)}_{!}1\right)\left[\text{dim}\left({ℱ}_{\overrightarrow{\nu }}\right)\right],\text{i.e.}\\ \phantom{LINE}\\ \begin{array}{ccccc}{D}_c^b\left({ℱ}_{\overrightarrow{\nu }}\right)& \stackrel{{\left({\pi }_{\overrightarrow{\nu }}\right)}_{!}}{⟶}& D_c^b\left(E_V\right)& \stackrel{\left[\text{dim}\left({ℱ}_{\overrightarrow{\nu }}\right)\right]}{\to }& D_c^b\left(E_V\right)\\ 1& ⟼& & ⟼& L_{\overrightarrow{\nu }}\end{array}\end{array}$$

where $1$ is the constant sheaf on ${ℱ}_{\overrightarrow{\nu }}$ and $\left[\text{dim}\left({ℱ}_{\overrightarrow{\nu }}\right)\right]$ is a shift, see IV (2.4).

Let $V$ be a $\Gamma \text{-graded}$ vector space of dimension $\nu$ and let $T$ and $W$ be $\Gamma \text{-graded}$ vector spaces such that $W\subseteq V$ and $T\cong V/W\text{.}$

1. Let $\overrightarrow{\tau }$ and $\overrightarrow{\omega }$ be partitions of $\text{dim}\left(T\right)$ and $\text{dim}\left(W\right),$ respectively. Then $${\text{Ind}}_{T,W}^V(L_{\overrightarrow{\tau }}\otimes L_{\overrightarrow{\omega }})=L_{\overrightarrow{\tau }\overrightarrow{\omega }},$$ where, if $\overrightarrow{\tau }=({\tau }^1,{\tau }^2,\dots ,{\tau }^s)$ and $\overrightarrow{\omega }=({\omega }^1,\dots ,{\omega }^t),$ then $\overrightarrow{\tau }\overrightarrow{\omega }=({\tau }^1,\dots ,{\tau }^s,{\omega }^1,\dots ,{\omega }^t)\text{.}$
2. Let $\overrightarrow{\nu }$ be a partition of $\text{dim}\left(V\right)\text{.}$ Then $${\text{Res}}_{T,W}^V{L}_{\overrightarrow{\nu }}\cong \underset{\overrightarrow{\tau },\overrightarrow{\omega }}{⨁}(L_{\overrightarrow{\tau }}\otimes L_{\overrightarrow{\omega }})\left[M^{\prime }(\overrightarrow{\tau },\overrightarrow{\omega })\right],$$ where the sum is over all $\overrightarrow{\tau },\overrightarrow{\omega }$ such that $\overrightarrow{\tau }$ is a partition of $\text{dim}\left(T\right),$ $\overrightarrow{\omega }$ is a partition of $\text{dim}\left(W\right)$ and $\overrightarrow{\tau }+\overrightarrow{\omega }=\overrightarrow{\nu }\text{.}$ The positive integer $M^{\prime }(\overrightarrow{\tau },\overrightarrow{\omega })$ is defined in (2.9) below.
3. Let $\nu ={\alpha }_i$ be a simple root for $𝔤$ and let $V$ be a $\Gamma \text{-graded}$ subspace such that $\text{dim}\left(V\right)={\alpha }_i\text{.}$ Define $L_i\in 𝒦\left({𝒬}_V\right)$ by $L_i=L_{\overrightarrow{\nu }}$ where $\overrightarrow{\nu }=\left({\alpha }_i\right)\text{.}$ Then $$\{\left[L_i\right],\left[L_i\right]\}=\frac{1}{1-q^2}\text{.}$$

The connection between $𝒦$ and the quantum group

We shall make the algebra

$$𝒦=\underset{\nu \in Q^{+}}{⨁}{𝒦}_{\nu }$$

a bit bigger by adding the $K_i^{\pm 1}\text{'s}$ that are in the quantum group $U_q𝔤\text{.}$ Let $𝔤$ be the finite dimensional complex simple Lie algebra corresponding to the Dynkin diagram given by $\Gamma$ and let $U_q𝔤$ be the rational version of the quantum group with $k=ℂ\left(q\right)$ where $q$ is an indeterminate. Let $U_q𝔥$ be the subalgebra of $U_q𝔤$ generated by $K_1^{\pm 1},\dots ,K_r^{\pm 1}\text{.}$ Let ${\alpha }_1,\dots ,{\alpha }_r$ be the simple roots corresponding to the Lie algebra $𝔤\text{.}$ Define

$$\begin{array}{c}\widetilde{𝒦}=\text{algebra generated by}𝒦\text{and}{K}_1^{\pm 1},\dots ,K_r^{\pm 1}\text{with the additional relations}\\ K_{i}x{K}_i^{-1}=q^{({\alpha }_i,\nu )}x,\text{for all}1\le i\le r\text{and all}x\in {𝒦}_{\nu },\end{array}$$

where the inner product in the exponent of $q$ is the inner product on ${𝔥}_{ℝ}^{*}$ given in II (2.7).

Define a map $j^{+}:𝒦\otimes 𝒦\to \widetilde{𝒦}\otimes \widetilde{𝒦}$ by

$$j^{+}(x\otimes y)=x{K}_1^{{\nu }_1}\dots K_r^{{\nu }_r}\otimes y,\text{if}x\in 𝒦\text{and}y\in {𝒦}_{\nu },\text{where}\nu ={\sum }_i{\nu }_i{\alpha }_i\text{.}$$

Use the map $j^{+}$ and the pseudo-comultiplication $r:𝒦\to 𝒦\otimes 𝒦$ defined in (2.5) to define a coproduct on $\widetilde{𝒦}$ by

$$\begin{array}{cccc}\Delta :& \widetilde{𝒦}& ⟶& \widetilde{𝒦}\otimes \widetilde{𝒦}\\ \\ & K_i^{\pm 1}& ⟼& K_i^{\pm 1}\otimes K_i^{\pm 1}& \text{for}1\le i\le r,\\ & x& ⟼& j^{+}r\left(x\right)& \text{for}x\in 𝒦,\end{array}$$

where $r:𝒦\to 𝒦\otimes 𝒦$ is the pseudo-comultiplication defined in (2.5). Then $\widetilde{𝒦}$ is a Hopf algebra!

Let $L_i$ be as defined in Theorem (2.7b). The algebra homomorphism determined by

$$\begin{array}{cccc}ℐ:& \widetilde{𝒦}& ⟶& U_q{𝔟}^{+}\\ \phantom{LINE}\\ & L_i& ⟼& E_i\\ & K_i^{\pm 1}& ⟼& K_i^{\pm 1}\end{array}$$

is an isomorphism of Hopf algebras.

Dictionary between $𝒦$ and $U_q{𝔟}^{+}$

Let us make a small dictionary between the algebra $𝒦$ and the quantum group $U_q{𝔟}^{+}\text{.}$ Our intent is to describe, conceptually, the correspondence between the structures inherent in the algebra $𝒦$ and the structures in the quantum group $U_q{𝔟}^{+}\text{.}$ The map $ℐ$ is the isomorphism given in Theorem (2.8).

$\widetilde{𝒦}$	is isomorphic to	$U_q{𝔟}^{+}\text{.}$
$\widetilde{𝒦}$ is the algebra generated by $𝒦$ and the $K_i^{\pm 1}\text{s.}$	Similarly,	$U_q{𝔟}^{+}$ is the algebra generated by $U_q{𝔫}^{+}$ and the $K_i^{\pm 1}\text{s.}$
$𝒦$ is graded, $𝒦=\underset{\nu \in Q^{+}}{⨁}{𝒦}_{\nu }\text{.}$	Similarly,	$U_q{𝔫}^{+}$ is graded, $U_n^{+}=\underset{\nu \in Q^{+}}{⨁}{\left(U_q{𝔫}^{+}\right)}_{\nu }\text{.}$
The shift functor $\left[n\right]$ gives rise to multiplication by $q^n$ in $𝒦$	which corresponds to	multiplication by $q^n$ in $U_q{𝔟}^{+}\text{.}$
The functor ${\text{Ind}}_{T,W}^V$	corresponds to	the multiplication in $U_q{𝔫}^{+}\text{.}$
The functor ${\text{Res}}_{T,W}^V$	corresponds to	the comultiplication in $U_q{𝔟}^{+}\text{.}$
The inner product $\{,\}$	corresponds to	the bilinear form $⟨,⟩$ pairing $U_q{𝔟}^{-}$ and $U_q{𝔟}^{+}\text{.}$
A partition $\overrightarrow{\nu }=({\nu }_1{\alpha }_{i_1},\dots ,{\nu }_l{\alpha }_{i_l})$ indexes $L_{\overrightarrow{\nu }}$	which maps, under $ℐ,$ to	$E_{i_1}^{\left({\nu }_1\right)}\dots E_{i_l}^{\left({\nu }_l\right)}$ where $E_i^{\left(n\right)}=E_i^n/\left[n\right]!\text{.}$
The Verdier duality functor D	corresponds to	the $ℂ\text{-algebra}$ involution ${}^{-}:U_q{𝔫}^{+}\to U_q{𝔫}^{+}$ which sends $q\mapsto q^{-1}$ and $E_i\mapsto E_i\text{.}$
The simple perverse sheaves in the various ${𝒬}_V$	map, under $ℐ,$ to	a canonical basis in $U_q{𝔫}^{+}\text{.}$

Definition of the constant $M^{\prime }(\tau ,\omega )$ which was used in (2.7)

Let $V$ be a $\Gamma \text{-graded}$ vector space and let $T$ and $W$ be $\Gamma \text{-graded}$ subspaces such that $W\subseteq V$ and $T\cong V/W\text{.}$ If $x\in E_V$ such that $xW\subseteq W$ then let $x_W$ be the linear transformation of $W$ induced by the action of $x$ on $W$ and let $x_T$ be the linear transformation of $T\cong V/W$ induced by the action of $x$ on $V/W\text{.}$ Let $\overrightarrow{\nu }$ be a partition of $\text{dim}\left(V\right)\text{.}$ If

$$f=(V=V^{\left(0\right)}\supseteq V^{\left(1\right)}\supseteq \dots \supseteq V^{\left(m\right)}=0)$$

is a flag of type $\overrightarrow{\nu }$ in $V$ then define

$$\begin{array}{ccc}{f}_W& =& ((V\cap W)=(V^{\left(0\right)}\cap W)\supseteq (V^{\left(1\right)}\cap W)\supseteq \dots \supseteq (V^{\left(m\right)}\cap W)=0)\text{and}\\ f_T& =& (p\left(V\right)=p\left(V^{\left(0\right)}\right)\supseteq p\left(V^{\left(1\right)}\right)\supseteq \dots \supseteq p\left(V^{\left(m\right)}\right)=0)\end{array}$$

where $p:V\to V/W$ is the canonical projection.

Let $\overrightarrow{\tau }$ be a partition of $\text{dim}\left(T\right)$ and let $\overrightarrow{\omega }$ be a partition of $\text{dim}\left(W\right),$ such that $\overrightarrow{\tau }+\overrightarrow{\omega }=\overrightarrow{\nu }\text{.}$

Define

$$\tilde{F}(\overrightarrow{\tau },\overrightarrow{\omega })=\{(x,f)\mid xW\subseteq W,f\text{is an}x\text{-stable flag of type}\overrightarrow{\nu }\text{in}V,\text{and}{f}_W\text{is a flag of type}\overrightarrow{\omega }W\}\text{.}$$

Define a map

$$\begin{array}{cccc}\alpha :& \tilde{F}(\overrightarrow{\tau },\overrightarrow{\omega })& ⟶& {ℱ}_{\overrightarrow{\tau }}\times {ℱ}_{\overrightarrow{\omega }}\\ & (x,f)& ⟼& ((x_T,f_T),(x_W,f_W))\end{array}$$

and define

$$M^{\prime }(\tau ,\omega )=\text{dim}\left(p_1\right)-\text{dim}\left(p_2\right)-2\text{dim}(G_V/P)+\text{dim}\left({ℱ}_{\overrightarrow{\nu }}\right)-\text{dim}\left({ℱ}_{\overrightarrow{\tau }}\right)-\text{dim}\left({ℱ}_{\overrightarrow{\omega }}\right)-2\text{dim}\left(\alpha \right)\text{.}$$

where $p_1$ and $p_2$ are the maps given in (2.4), $P$ is the parabolic subgroup of $G_V$ defined in (2.4), and $\text{dim}\left(p_1\right),$ $\text{dim}\left(p_2\right)$ and $\text{dim}\left(\alpha \right)$ are the dimensions of the fibers of the maps $p_1,$ $p_2,$ and $\alpha ,$ respectively.

Definition of the vector spaces ${\mathscr{H}}^{j+2\text{dim}(G\backslash \Omega )}\left(u_{!}(t_{♭}{s}^{*}{B}_1\otimes t_{♭}{s}^{*}{B}_2)\right)$ from (2.6)

Let $\Omega$ be a smooth irreducible algebraic variety with a free action of $G_V$ such that the $\overline{{\mathbb{Q}}_l}\text{-cohomology}$ of $\Omega$ is zero in degrees $1,2,\dots ,m$ where $m$ is a large integer. Consider the diagram

$$\begin{array}{ccc}\begin{array}{ccccc}{E}_V& \stackrel{s}{⟵}& \Omega \times E_V& \stackrel{t}{⟶}& G\backslash (\Omega \times E_V)\\ x& ⟼ & (\omega ,x)& ⟼& G_V(\omega ,x)\end{array}& \text{and the diagram}& \begin{array}{ccc}{G}_V\backslash (\Omega \times E_V)& \stackrel{u}{⟶}& \left\{\text{point}\right\}\text{.}\end{array}\end{array}$$

These diagrams induce diagrams

$$\begin{array}{c}\begin{array}{ccccc}{D}_c^b\left(E_V\right)& \stackrel{s^{*}}{⟶}& D_c^b(\Omega \times E_V)& \stackrel{t_{♭}}{⟶}& D_c^b(G\backslash (\Omega \times E_V))\end{array}\text{and}\\ \begin{array}{ccc}{D}_c^b(G\backslash (\Omega \times E_V))& \stackrel{u_{!}}{⟶}& D_c^b\left(\left\{\text{point}\right\}\right)\text{.}\end{array}\end{array}$$

With these notations one has that ${\mathscr{H}}^{j+2\text{dim}(G\backslash \Omega )}\left(u_{!}(t_{♭}{s}^{*}{B}_1\otimes t_{♭}{s}^{*}{B}_2)\right)$ is a sheaf on the space $\left\{\text{point}\right\},$ i.e. a $\overline{{\mathbb{Q}}_l}\text{-vector}$ space.

Some remarks on Part II of Lusztig's book

The construction of the algebra $𝒦$ and the relationship between it and the quantum group is detailed in Lusztig's book [Lus1993]. Lusztig works in much more generality there

1. Lusztig allows $\Gamma$ to be an arbitrary quiver, rather than just a quiver gotten by orienting a Dynkin diagram of type $A,$ $D$ or $E\text{.}$ It does not require any more theory than what we have already outlined in order to define the algebra $𝒦$ in this more general setting.
2. Lusztig wants to construct algebras $𝒦$ which will be isomorphic to the nonnegative parts of the quantum groups corresponding to general Dynkin diagrams. In order to do this he must first consider only diagrams with single bonds and then 'fold' the diagram by analyzing the action of an automorphism of the diagram. The addition of the folding automorphism into the theory is a nontrivial extension of what we have developed in these notes.
3. We have ignored the effect of the orientation of the quiver. If one wants to compare the algebras $𝒦$ that are obtained by orienting the same quiver in different ways one must analyze a Fourier-Deligne transform between these two different algebras. The amazing thing is that, after one extends the algebras by adding the $K_i^{\pm 1}\text{s}$ that are in the quantum group, the two different algebras (from the different orientations) become isomorphic!

The connection between representations of quivers and perverse sheaves

Correspondence between orbits and isomorphism classes of representations of $\Gamma$

Let $\Gamma$ be a quiver obtained by orienting the edges of a Dynkin diagram of type $A,$ $D$ or $E\text{.}$ For convenience we label the vertices by $1,2,\dots ,r\text{.}$ Let $𝔤$ be the finite dimensional complex simple Lie algebra corresponding to the Dynkin diagram given by $\Gamma \text{.}$

Let $p$ be a positive prime integer and let $\overline{{𝔽}_p}$ be the algebraic closure of the finite field ${𝔽}_p$ with $p$ elements. Fix an element $\nu \in Q^{+}$ (see VII (1.2)) and a $\Gamma \text{-graded}$ vector space $V$ over $\overline{{𝔽}_p}$ such that $\text{dim}\left(V\right)=\nu \text{.}$ Define

$$G_V=\prod _{i}GL\left(V_i\right)\text{and}{E}_V=\underset{i\to j}{⨁}\text{Hom}(V_i,V_j),$$

where the sum in the definition of $E_V$ is over all edges of $\Gamma \text{.}$ The natural action of $G_V$ on $E_V$ is given by

$$g·\left({\varphi }_{ij}\right)=\left(g_j{\varphi }_{ij}{g}_i^{-1}\right),\text{if}\left({\varphi }_{ij}\right)\in E_V\text{and}g=(g_1,\dots ,g_r)\in G_V\text{.}$$

The group $G_V$ is an algebraic group over $\overline{{𝔽}_p}$ and $E_V$ is a variety over $\overline{{𝔽}_p}$ with a $G_V$ action. Each element $\left({\varphi }_{ij}\right)\in E_V$ determines a representation of $\Gamma$ of dimension $\text{dim}\left(V\right)\text{.}$ Each $G_V\text{-orbit}$ in $E_V$ determines an isomorphism class of representations of $\Gamma \text{.}$ Let us make this correspondence precise.

An orbit index for $V$ is a sequence of positive integers labeled by the positive roots

$$\overrightarrow{c}={\left(c_{\alpha }\right)}_{\alpha \in R^{+}}\text{such that}\sum _{\alpha \in R^{+}}{c}_{\alpha }\alpha =\text{dim}\left(V\right),$$

where $R^{+}$ is the set of positive roots for $𝔤\text{.}$ For each orbit index $\overrightarrow{c}$ for $V$ define a representation of $\Gamma$ by

$$R_{\overrightarrow{c}}=\underset{\alpha \in R^{+}}{⨁}{e}_{\alpha }^{\oplus c_{\alpha }}\text{and let}{𝒪}_{\overrightarrow{c}}=\text{the}{G}_V\text{-orbit in}{E}_V\text{corresponding to}{R}_{\overrightarrow{c}},$$

where $e_{\alpha }$ is the indecomposable representation of $\Gamma$ indexed by the positive root $\alpha ,$ see Theorem (1.2b). Then we have a one-to-one correspondence

$$\begin{array}{ccc}{G}_V\text{orbits in}{E}_V& \stackrel{1-1}{⟷}& \text{isomorphism classes of representations of}\Gamma \text{of dimension}\nu \\ {𝒪}_{\overrightarrow{c}}& ⟷& \left[R_{\overrightarrow{c}}\right]\end{array}$$

Realizing the structure constants of the Hall algebra in terms of orbits

Let $q$ be a power of the prime $p\text{.}$ Since $E_V$ is a variety of $\overline{{𝔽}_p}$ there is an action of the $q\text{th}$ power Frobenius map $F$ on $E_V,$ see [Car1985] p. 503. If $X$ is a subset of $E_V$ then let $X^F$ denote the set of points of $X$ which are fixed under the action of the Frobenius map $F\text{.}$

Let $T$ and $W$ be $\Gamma \text{-graded}$ vector spaces such that $W\subseteq V$ and $T\cong V/W\text{.}$ Recall the diagram

$$\begin{array}{ccccccc}{E}_T\times E_W& \stackrel{p_1}{⟵}& G_V{\times }_U𝒮& \stackrel{p_2}{⟶}& G_V{\times }_P𝒮& \stackrel{p_3}{⟶}& E_V\\ (x_T,x_W)& ⟼ & (g,x)& ⟼& (g,x)& ⟼& gx\end{array}$$

given in (2.4). Let $\overrightarrow{a},$ $\overrightarrow{b},$ and $\overrightarrow{c}$ be orbits indices for $T,$ $W$ and $V,$ respectively. Then we have

$$\begin{array}{ccccccc}{E}_T\times E_W& \stackrel{p_1}{⟵}& G_V{\times }_U𝒮& \stackrel{p_2}{⟶}& G_V{\times }_P𝒮& \stackrel{p_3}{⟶}& E_V\\ {𝒪}_{\overrightarrow{a}}\times {𝒪}_{\overrightarrow{b}}& ⟷& p_1^{-1}({𝒪}_{\overrightarrow{a}}\times {𝒪}_{\overrightarrow{b}})& ⟼& p_2\left(p_1^{-1}({𝒪}_{\overrightarrow{a}}\times {𝒪}_{\overrightarrow{b}})\right)\\ & & & & p_3^{-1}\left({𝒪}_{\overrightarrow{c}}\right)& ⟷& {𝒪}_{\overrightarrow{c}}\end{array}$$

Let $M=R_{\overrightarrow{a}},$ $N=R_{\overrightarrow{b}}$ and $P=R_{\overrightarrow{c}}$ be the representations of $\Gamma$ given in (3.1). By a direct count, we have

$$C_{M,N}^P=\text{Card}\left({(p_2\left(p_1^{-1}({𝒪}_{\overrightarrow{a}}\times {𝒪}_{\overrightarrow{b}})\right)\bigcap p_3^{-1}\left({𝒪}_{\overrightarrow{c}}\right))}^F\right)\text{.}$$

where $C_{M,N}^P$ are the structure coefficients of the Hall algebra $R\Gamma$ given in (1.3).

Rewriting the Hall algebra in terms of functions constant on orbits

Let $q$ be a power of the prime $p\text{.}$ On any variety $Y$ over $\overline{{𝔽}_p}$ there is an action of the $q\text{th}$ power Frobenius map $F$ on $E_V,$ see [Car1985] p. 503. If $X$ is a subset of $Y$ then $X^F$ denotes the set of points of $X$ which are fixed under the action of the Frobenius map $F\text{.}$

Let $l$ be a positive prime number, invertible in $\overline{{𝔽}_p}\text{.}$ Let $\overline{{\mathbb{Q}}_l}$ be the algebraic closure of the field of $l\text{-adic}$ numbers. Define

$$\begin{array}{ccc}{K}_{\nu }& =& \text{the vector space of}\overline{{\mathbb{Q}}_l}\text{-valued functions on}{\left(E_V\right)}^F\text{which are constant on the orbits}$ {\left(𝒪\overrightarrow{c}\right)}^F$\text{for all orbit indexes}\overrightarrow{c}\text{for}V\text{.}\end{array}$$

Define

$$K=\underset{\nu \in Q^{+}}{⨁}{K}_{\nu },$$

where $Q^{+}$ is as in VII (1.2).

Define a multiplication on $K$ as follows. Let $T$ and $W$ be $\Gamma \text{-graded}$ vector spaces such that $W\subseteq V$ and $T\cong V/W\text{.}$ Recall the diagram

$$\begin{array}{ccccccc}{E}_T\times E_W& \stackrel{p_1}{⟵}& G_V{\times }_U𝒮& \stackrel{p_2}{⟶}& G_V{\times }_P𝒮& \stackrel{p_3}{⟶}& E_V\\ (x_T,x_W)& ⟼ & (g,x)& ⟼& (g,x)& ⟼& gx\end{array}$$

given in (2.4). Let $\tau =\text{dim}\left(T\right)$ and $\omega =\text{dim}\left(W\right)\text{.}$ Given $f_1\in K_{\tau }$ and $f_2\in K_{\omega }$ define a function $f_1*f_2$ as follows:
If $x\in {\left(E_V\right)}^F$ then

$$(f_1*f_2)\left(x\right)=\sum _{x_T,x_W}{C}_{T,W}^V{f}_1\left(x_T\right)f_2\left(x_W\right),$$

where the sum is over all $x_T\in {\left(E_T\right)}^F$ and $x_W\in {\left(E_W\right)}^F,$ and

$$\begin{array}{ccc}{C}_{T,W}^V& =& \frac{\text{Card}\left(\{(y,f)\in {\left(G_V{\times }_P𝒮\right)}^F\mid p_1(y,f)=(x_T,x_W),p_3\left(p_2(y,f)\right)=x\}\right)}{\text{Card}\left({\left(G_T\right)}^F\right)\text{Card}\left({\left(G_W\right)}^F\right)}\end{array}$$

Let $\overrightarrow{c}$ be an orbit index and let ${\chi }_{\overrightarrow{c}}$ be the characteristic function of the orbit ${𝒪}_{\overrightarrow{c}},$ i.e.

$$\text{for}x\in {\left(E_V\right)}^F,{\chi }_{\overrightarrow{c}}\left(x\right)=\{\begin{array}{cc}1,& \text{if}x\in {\left({𝒪}_{\overrightarrow{c}}\right)}^F,\\ 0,& \text{otherwise.}\end{array}$$

Then it follows from the observation in (3.2) that the map

$$\begin{array}{ccc}K& ⟶& R\Gamma \\ {\chi }_{\overrightarrow{c}}& ⟼& \left[R_{\overrightarrow{c}}\right]\end{array}$$

is an isomorphism of algebras, where $R\Gamma$ is the Hall algebra defined in (1.3).

The isomorphism between $𝒦$ and $K$

Let $\overrightarrow{a}$ be an orbit index and let ${𝒪}_{\overrightarrow{a}}$ be the corresponding $G_V\text{-orbit}$ in $E_V$ as defined in (3.1). Let $F_{\overrightarrow{c}}$ be the constant sheaf $\overline{{\mathbb{Q}}_l}$ on the orbit ${𝒪}_{\overrightarrow{c}}$ extended by 0 on the complement. This sheaf can be viewed as the complex of sheaves $A,$ for which $A^0=F_{\overrightarrow{c}}$ and $A^i=0,$ for all $i\ne 0\text{.}$ In this way $F_{\overrightarrow{c}}$ can be viewed as an element of ${𝒬}_V,$ see IV (1.4), and the isomorphism class $\left[F_{\overrightarrow{c}}\right]$ of $F_{\overrightarrow{c}}$ is an element of $𝒦\text{.}$

Let $𝒦$ be the algebra defined in §2 and let $K$ be the algebra defined in (3.3). For each orbit index $\overrightarrow{c}$ let ${𝒪}_{\overrightarrow{c}}$ be the corresponding $G_V$ orbit in $E_V,$ as given in (3.1), and let ${\chi }_{\overrightarrow{c}}$ be the characteristic function of the orbit ${𝒪}_{\overrightarrow{c}}\text{.}$ The map

$$\begin{array}{ccc}𝒦& ⟶& K\\ \left[F_{\overrightarrow{c}}\right]& ⟼& {\chi }_{\overrightarrow{c}}\end{array}$$

is an isomorphism of algebras.

This theorem is a consequence of an analogue of the Grothendieck trace formula. The Grothendieck trace formula, [Car1985] p. 504, is the formula

$$\begin{array}{ccc}\mid X^F\mid & =& \sum _{i=0}^{2\text{dim}\left(X\right)}{(-1)}^i\text{Tr}(F,H_c^i(X,{\mathbb{Q}}_l)),\end{array}$$

which describes the number of points of $X$ which are fixed under a Frobenius map $F$ in terms of the trace of the action of the Frobenius map on the $l\text{-adic}$ cohomology $H_c^i(X,{\mathbb{Q}}_l)$ of the variety $X\text{.}$

Theorems (3.4) and (2.8) together show that there is a natural connection between the algebra $𝒦$ and the Hall algebra $R\Gamma$ which was introduced in (1.3).

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.

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