Hall algebras of curves, commuting varieties and Langlands duality.(English)Zbl 1252.14012
The authors construct an isomorphism between the (universal) spherical Hall algebra of a smooth projective curve of genus \(g\) and a convolution algebra in the (equivariant) \(K\)-theory of the genus \(g\) commuting varieties \({C_{\mathfrak{gl}_r}=\big\{ (x_1, \dots, x_g, y_1, \dots, y_g) \in \mathfrak{gl}_r^{2g} \mid \sum_{i=1}^g [x_i,y_i]=0\big\}}\). This isomorphism can be viewed as a version of the geometric Langlands duality in the formal neighborhood of the trivial local system, for the group \(\mathrm{GL} _{r }\). This is extended to all reductive groups. Under this correspondence the image of the skyscraper sheaf supported on the trivial local system is computed.
Let \(X\) be a smooth connected projective curve of genus \(g\) defined over a finite field \(\mathbb F_q\). Let \(\mathcal C\!oh(X)\) be the category of coherent sheaves over \(X\), let \(\mathcal T\!or(X)\) be the full subcategory of torsion sheaves. Let \(\mathcal I\) be the set of isomorphism classes of coherent sheaves on \(X\). Consider the vector space \(\mathbf{H}_X\) of all functions \(f: \mathcal I\to \mathbb C\) with finite support. For a sheaf \(\mathcal F\) let \(r_{\mathcal F}\) and \(d_{\mathcal F}\) be its rank and degree respectively. For two sheaves \(\mathcal F\) and \(\mathcal G\) define the Euler form by \[ \langle\mathcal F, \mathcal G\rangle = \dim \operatorname{Hom}(\mathcal F, \mathcal G)-\dim \mathrm{Ext}^1(\mathcal F, \mathcal G)=(1-g)r_{\mathcal F}r_{\mathcal G}+(r_{\mathcal F}d_{\mathcal G}-r_{\mathcal G}d_{\mathcal F}). \] Choose a square root \(v\) of \(q^{-1}\). Then one defines on \(\mathbf{H}_X\) the multiplication \[ (f\cdot g)(\mathcal R)=\sum\limits_{\mathcal N\subseteq \mathcal R} v^{-\langle \mathcal R/\mathcal N, \mathcal N\rangle} f(\mathcal R/\mathcal N)g(\mathcal N). \] For \(d\in \mathbb Z\) let \(\mathrm{Pic}^d X\) be the set of isomorphism classes of line bundles on \(X\) of degree \(d\). For a line bundle on \(X\) let \(1_{\mathcal L}\) be the characteristic function of \(\mathcal L\), let \[ 1^{ss}_{1, d}=\sum\limits_{\mathcal L\in \mathrm{Pic}^d(X)} 1_{\mathcal L}. \] Denote by \(\mathbf{U}^>_X\) the subalgebra of \(\mathbf{H}_X\) generated by \(\{1_{1, d}^{ss}\}_{d\in \mathbb Z}\). The authors identify \(\mathbf{U}^>_X\) with a certain shuffle algebra \(\mathbf{A}\) in the space of symmetric polynomials.
For a fixed genus \(g\), let \(GSP\) be the group of the symplectic similitudes of \(\mathbb C^{2g}\) with the canonical symplectic form. Let \(R\) be the complexified representation ring of \(GSP\). Let \(K\) be its fraction field. Using the presentation of \(\mathbf{U}^>_X\) as a shuffle algebra and substituting the field \(\mathbb C\) by \(K\) the authors define the \(K\)-algebra \(\mathbf{A}_K\) and put \(\mathbf{U}^>_K=\text\textbf{A}_K\). Then \(\mathbf{A}_R\) is a torsion-free intergal form of \(\mathbf{A}_K\), i. e., an \(R\)-subalgebra of \(\mathbf{A}_K\) such that \(\mathbf{A}_R\otimes_R K=\text\textbf{A}_K\). For every fixed curve \(X\) of genus \(g\) there exists a natural specialization map \[ \mathbf{A}_R\to \text\textbf{U}^>_X. \] Hence one can put \(\mathbf{U}^>_R=\text\textbf{A}_R\) and call it the universal spherical Hall algebra of a fixed genus \(g\). Modifying the multiplication on \(\mathbf{U}^>_X\) using the tensoring by powers of the canonical bundle on \(X\) one can obtain its twisted version \(\dot{\mathbf{ {U}}}^>_X\) with associative multiplication. This also gives a presentation of \(\dot{\mathbf{ {U}}}^>_X\) as a shuffle algebra and hence there is its universal version \(\dot{\mathbf{ {U}}}^>_R\).
One constructs the spherical \(K\)-theoretic Hall algebra \(\bar{\mathbf{C}}\) as follows. Consider \[C_{\mathfrak{gl}_r}=\big\{ (x_1, \dots, x_g, y_1, \dots, y_g) \in \mathfrak{gl}_r^{2g} \mid \sum_{i=1}^g [x_i,y_i]=0\big\}.\] Since \(\mathrm{GL}_r\) acts by the adjoint action on \(\mathfrak{gl}_r\), there is an action of \(GSP\times \mathrm{GL}_r\) on \(C_{\mathfrak{gl}_r}\). Let \(K^{GSP\times \mathrm{GL}_r}(C_{\mathfrak{gl}_r})\) be the complexified Grothendieck group of the abelian category of the \((GSP\times \mathrm{GL}_r)\)-equivariant coherent sheaves over \(C_{\mathfrak{gl}_r}\). Then the convolution product in equivariant \(K\)-theory equips the graded space \[ \mathbf{C}= \mathbb C\cdot 1 \oplus \bigoplus_{r\geq 1} K^{GSP\times \mathrm{GL}_r}(C_{\mathfrak{gl}_r})=\bigoplus_{r\geq 0} \text\textbf{C}_r, \] with the structure of an associative \(R\)-algebra with \(1\). It’s subalgebra generated by elements of degree \(1\), i. e., by \(\mathbf{C}_1=K^{GSP\times \mathrm{GL}_1}(C_{\mathfrak{gl}_1})=K^{GSP\times \mathrm{GL}_1}(\mathbb C^{2g})\), is denoted by \(\mathbf{C}'\).
The inclusion \(C_{\mathfrak{gl}_r}\subset \mathfrak{gl}_r^g\times \mathfrak{gl}_r^g\) and the direct image construction induce a map \(\mathbf{C}_r\to K^{GSP\times \mathrm{GL}_r}(\mathfrak{gl}_r^g\times \mathfrak{gl}_r^g)\). The authors conjecture that this map is injective (cf. Conjecture 2.6). Let \(\bar{\mathbf{C}}_r\) be the image of \(\mathbf{C}'_r=\text\textbf{C}_r\cap \text\textbf{C}'\), let \(\bar{\mathbf{C}}=\bigoplus_{r\geq 0} \bar{\mathbf{C}}_r\), then there is a surjective \(R\)-algebra homomorphism \(\mathbf{C}'\to \bar{\text\textbf{C}}\), which is just an isomorphism if Conjecture 2.6 mentioned above is true.
The main result of the paper, Theorem 3.1, is a construction of an \(R\)-algebra anti-isomorphism \(\Theta_R: \bar{\mathbf{C}}\to \dot{\mathbf{ {U}}}^>_R\).
The paper under review consists of an introduction, 3 sections and two appendices. Section 1 deals with Hall algebras of curves. Section 2 is devoted to the Hall algebra in the \(K\)-theory of the genus \(g\) commuting varieties. In Section 3 the main result, i. e., the Langlands isomorphism, Theorem 3.1, is stated. The extensions of the results of the paper to the case of principal Hall algebra and to the case of arbitrary reductive groups are sketched in Appendix A and Appendix B respectively.
Let \(X\) be a smooth connected projective curve of genus \(g\) defined over a finite field \(\mathbb F_q\). Let \(\mathcal C\!oh(X)\) be the category of coherent sheaves over \(X\), let \(\mathcal T\!or(X)\) be the full subcategory of torsion sheaves. Let \(\mathcal I\) be the set of isomorphism classes of coherent sheaves on \(X\). Consider the vector space \(\mathbf{H}_X\) of all functions \(f: \mathcal I\to \mathbb C\) with finite support. For a sheaf \(\mathcal F\) let \(r_{\mathcal F}\) and \(d_{\mathcal F}\) be its rank and degree respectively. For two sheaves \(\mathcal F\) and \(\mathcal G\) define the Euler form by \[ \langle\mathcal F, \mathcal G\rangle = \dim \operatorname{Hom}(\mathcal F, \mathcal G)-\dim \mathrm{Ext}^1(\mathcal F, \mathcal G)=(1-g)r_{\mathcal F}r_{\mathcal G}+(r_{\mathcal F}d_{\mathcal G}-r_{\mathcal G}d_{\mathcal F}). \] Choose a square root \(v\) of \(q^{-1}\). Then one defines on \(\mathbf{H}_X\) the multiplication \[ (f\cdot g)(\mathcal R)=\sum\limits_{\mathcal N\subseteq \mathcal R} v^{-\langle \mathcal R/\mathcal N, \mathcal N\rangle} f(\mathcal R/\mathcal N)g(\mathcal N). \] For \(d\in \mathbb Z\) let \(\mathrm{Pic}^d X\) be the set of isomorphism classes of line bundles on \(X\) of degree \(d\). For a line bundle on \(X\) let \(1_{\mathcal L}\) be the characteristic function of \(\mathcal L\), let \[ 1^{ss}_{1, d}=\sum\limits_{\mathcal L\in \mathrm{Pic}^d(X)} 1_{\mathcal L}. \] Denote by \(\mathbf{U}^>_X\) the subalgebra of \(\mathbf{H}_X\) generated by \(\{1_{1, d}^{ss}\}_{d\in \mathbb Z}\). The authors identify \(\mathbf{U}^>_X\) with a certain shuffle algebra \(\mathbf{A}\) in the space of symmetric polynomials.
For a fixed genus \(g\), let \(GSP\) be the group of the symplectic similitudes of \(\mathbb C^{2g}\) with the canonical symplectic form. Let \(R\) be the complexified representation ring of \(GSP\). Let \(K\) be its fraction field. Using the presentation of \(\mathbf{U}^>_X\) as a shuffle algebra and substituting the field \(\mathbb C\) by \(K\) the authors define the \(K\)-algebra \(\mathbf{A}_K\) and put \(\mathbf{U}^>_K=\text\textbf{A}_K\). Then \(\mathbf{A}_R\) is a torsion-free intergal form of \(\mathbf{A}_K\), i. e., an \(R\)-subalgebra of \(\mathbf{A}_K\) such that \(\mathbf{A}_R\otimes_R K=\text\textbf{A}_K\). For every fixed curve \(X\) of genus \(g\) there exists a natural specialization map \[ \mathbf{A}_R\to \text\textbf{U}^>_X. \] Hence one can put \(\mathbf{U}^>_R=\text\textbf{A}_R\) and call it the universal spherical Hall algebra of a fixed genus \(g\). Modifying the multiplication on \(\mathbf{U}^>_X\) using the tensoring by powers of the canonical bundle on \(X\) one can obtain its twisted version \(\dot{\mathbf{ {U}}}^>_X\) with associative multiplication. This also gives a presentation of \(\dot{\mathbf{ {U}}}^>_X\) as a shuffle algebra and hence there is its universal version \(\dot{\mathbf{ {U}}}^>_R\).
One constructs the spherical \(K\)-theoretic Hall algebra \(\bar{\mathbf{C}}\) as follows. Consider \[C_{\mathfrak{gl}_r}=\big\{ (x_1, \dots, x_g, y_1, \dots, y_g) \in \mathfrak{gl}_r^{2g} \mid \sum_{i=1}^g [x_i,y_i]=0\big\}.\] Since \(\mathrm{GL}_r\) acts by the adjoint action on \(\mathfrak{gl}_r\), there is an action of \(GSP\times \mathrm{GL}_r\) on \(C_{\mathfrak{gl}_r}\). Let \(K^{GSP\times \mathrm{GL}_r}(C_{\mathfrak{gl}_r})\) be the complexified Grothendieck group of the abelian category of the \((GSP\times \mathrm{GL}_r)\)-equivariant coherent sheaves over \(C_{\mathfrak{gl}_r}\). Then the convolution product in equivariant \(K\)-theory equips the graded space \[ \mathbf{C}= \mathbb C\cdot 1 \oplus \bigoplus_{r\geq 1} K^{GSP\times \mathrm{GL}_r}(C_{\mathfrak{gl}_r})=\bigoplus_{r\geq 0} \text\textbf{C}_r, \] with the structure of an associative \(R\)-algebra with \(1\). It’s subalgebra generated by elements of degree \(1\), i. e., by \(\mathbf{C}_1=K^{GSP\times \mathrm{GL}_1}(C_{\mathfrak{gl}_1})=K^{GSP\times \mathrm{GL}_1}(\mathbb C^{2g})\), is denoted by \(\mathbf{C}'\).
The inclusion \(C_{\mathfrak{gl}_r}\subset \mathfrak{gl}_r^g\times \mathfrak{gl}_r^g\) and the direct image construction induce a map \(\mathbf{C}_r\to K^{GSP\times \mathrm{GL}_r}(\mathfrak{gl}_r^g\times \mathfrak{gl}_r^g)\). The authors conjecture that this map is injective (cf. Conjecture 2.6). Let \(\bar{\mathbf{C}}_r\) be the image of \(\mathbf{C}'_r=\text\textbf{C}_r\cap \text\textbf{C}'\), let \(\bar{\mathbf{C}}=\bigoplus_{r\geq 0} \bar{\mathbf{C}}_r\), then there is a surjective \(R\)-algebra homomorphism \(\mathbf{C}'\to \bar{\text\textbf{C}}\), which is just an isomorphism if Conjecture 2.6 mentioned above is true.
The main result of the paper, Theorem 3.1, is a construction of an \(R\)-algebra anti-isomorphism \(\Theta_R: \bar{\mathbf{C}}\to \dot{\mathbf{ {U}}}^>_R\).
The paper under review consists of an introduction, 3 sections and two appendices. Section 1 deals with Hall algebras of curves. Section 2 is devoted to the Hall algebra in the \(K\)-theory of the genus \(g\) commuting varieties. In Section 3 the main result, i. e., the Langlands isomorphism, Theorem 3.1, is stated. The extensions of the results of the paper to the case of principal Hall algebra and to the case of arbitrary reductive groups are sketched in Appendix A and Appendix B respectively.
Reviewer: Oleksandr Iena (Luxembourg)
MSC:
| 14D24 | Geometric Langlands program (algebro-geometric aspects) |
| 22E57 | Geometric Langlands program: representation-theoretic aspects |
| 16S99 | Associative rings and algebras arising under various constructions |
| 19E99 | \(K\)-theory in geometry |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
Hall algebra;spherical Hall algebra;\(K\)-theory;Langlands duality;Langlands program;trivial local system;Hecke operators;finite field;shuffle algebraCite
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