Graded $R_{\alpha }$-modules

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

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 2 March 2010

Graded $R_{\alpha }$-modules

A $\mathbb{Z}$-graded vector space is a vector space with a decomposition $$V={\displaystyle \underset{i\in \mathbb{Z}}{\oplus }{V}_i\text{and}\mathrm{gdim}\left(V\right)=\sum _{i\in \mathbb{Z}}{q}^i\dim \left(V\right)}$$ is the graded dimension of $V$. If $M$ is a $\mathbb{Z}$-graded $R_{\alpha }$-module then, as graded vector spaces, $$M={\displaystyle \underset{u\in {\Gamma }^{\alpha }}{\oplus }{e}_{u}M\text{and}\mathrm{gdim}\left(M\right)=\sum _{u\in {\Gamma }^{\alpha }}\mathrm{gdim}\left(e_{u}M\right)f_u}$$ is the graded character of $M$.

Let $\alpha ,\beta \in Q^{+}$ and let $k$ and $l$ be the lengths of the words in ${\Gamma }^{\alpha }$ and ${\Gamma }^{\beta }$, respectively. Then $${\displaystyle {\Gamma }^{\alpha +\beta }=\underset{\sigma \in S_{k+l}/\left(S_k\times S_l\right)}{⨆}\text{and}\begin{array}{rcl}{R}_{\alpha }\otimes R_{\beta }& \mapsto & R_{\alpha +\beta }\\ e_u\otimes e_v& \mapsto & e_{uv}\\ x_i{e}_u\otimes e_v& \mapsto & x_i{e}_{uv}\\ {\tau }_i{e}_u\otimes e_v& \mapsto & {\tau }_i{e}_{uv}\\ e_u\otimes x_j{e}_v& \mapsto & x_{j+k}{e}_{uv}\\ e_u\otimes {\tau }_j{e}_v& \mapsto & {\tau }_{j+k}{e}_{uv}\end{array}}$$ defines an injection (of nonunital algebras).

[KL1, Prop 2.16] As a right $R_{\alpha }\otimes R_{\beta }$-module, $R_{\alpha +\beta }$ has basis $$\left\{{\tau }_{\sigma }{1}_{\alpha \beta }|\sigma \in S_{k+l}/\left(S_k\times S_l\right)\right\},\text{where}{1}_{\alpha \beta }=\sum _{u\in {\Gamma }^{\alpha },v\in {\Gamma }^{\beta }}{e}_{uv},$$ and, for each minimal length representative $\sigma$ of a coset in $S_{k+l}/\left(S_k\times S_l\right),$ we fix a reduced word $$\sigma =s_{i_1}\dots s_{i_l}\text{and set}{\tau }_{\sigma }={\tau }_{i_1}\dots {\tau }_{i_l}.$$

Let $R_{\alpha }$-mod be the category of $\mathbb{Z}$-graded $R_{\alpha }$-modules. For $M\in R_{\alpha }$-mod and $N\in R_{\beta }$ define $$M\circ N={\mathrm{Ind}}_{R_{\alpha }\otimes R_{\beta }}^{R_{\alpha +\beta }}\left(M\otimes N\right).$$

Let $K\left(R_{\alpha }\text{-mod}\right)$ be the Grothendieck group of $\mathbb{Z}$-graded $R$-modules and define a $$\text{product on}\underset{\alpha \in Q^{+}}{⨁}K\left(R_{\alpha }\text{-mod}\right)\text{by}\left[M\right].\left[N\right]=\left[M\circ N\right].$$

The following theorem says that the categories $R_{\alpha }$-mod form a categorification of $U_q{𝔫}^{-}.$ It is the main theorem of this theory.

[KL, Prop 3.4] The graded character map $$\begin{array}{rcll}\mathrm{gch}:& \underset{\alpha \in Q^{+}}{⨁}K\left(R_{\alpha }\text{-mod}\right)& \to & U_q{𝔫}^{-}\\ & \left[M\right]& \mapsto & \mathrm{gch}\left[M\right]\end{array}$$ is an algebra ismorphism.

1. If $M\in R_{\alpha }$-mod then $\mathrm{gch}\left(M\right)\in U_q{𝔫}^{-}.$
2. If $M\in R_{\alpha }$-mod and $N\in R_{\beta }$-mod then $$\mathrm{gch}\left({\mathrm{Ind}}_{R_{\alpha }\otimes R_{\beta }}^{R_{\alpha +\beta }}\left(M\otimes N\right)\right)=\mathrm{gch}\left(M\right)\bigsqcup \bigsqcup \mathrm{gch}\left(N\right).$$

Proof

1. (sketch only) $$\mathrm{gch}\left(M\right)=\sum _{u\in {\Gamma }^{\alpha }}\mathrm{gdim}\left(e_{u}M\right)f_u=\sum _{u\in \stackrel{\alpha }{\Gamma }}\mathrm{gdim}\left(\mathrm{Hom}\left(M,R_{\alpha }{e}_u\right)\right)f_u,$$ and so by Roquier's complex [Ro, Lemma 3.13], $\mathrm{gch}\left(M\right)$ satisfies Serre relations. Thus, by Proposition ?, $\mathrm{gch}\left(M\right)\in U_q{𝔫}^{-}.$
1. (second proof) (sketch) By understanding the intertwiners thoroughly, analyse the $R_{\alpha }$-modules when $\alpha =-⟨{\alpha }_i,{\alpha }_j^{\vee }⟩{\alpha }_1+{\alpha }_2.$ $$1112\leftrightarrow 11121\leftrightarrow 11211\leftrightarrow 12111\leftrightarrow 21111$$
2. This follows from the fact that $\left\{{\tau }_{\sigma }{1}_{\alpha \beta }|\sigma \in S_{k+l}/S_k\times S_l\right\}$ is a basis of $R_{\alpha +\beta }$ as a right $R_{\alpha }\otimes R_{\beta }$-module.

References

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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