Hall Algebra Modules, Fock Spaces, and Geometric Actions

When $v=0$, both maps have zero source or target and hence are zero. There is one solution, stability is vacuous, and $G_0$ is trivial, so $\mathfrak{M}(0,1)$ is a point. Its invariant coordinate ring is $\mathbb{C}$, whose only prime ideal is $(0)$; hence ${\mathfrak{M}}_0(0,1)$ and the fibre $\mathcal{L}(0,1)$ are points by [F2]–[F3].

When $v=1$, write $I(w)=iw$ and $J(z)=jz$ for $i,j\in \mathbb{C}$. The moment equation is $ij=0$ and stability is $j\mathrm{\ne }0$, so $i=0$ on the stable locus. Since $G_1={\mathbb{C}}^{\times }$ sends $j$ to $j{g}^{-1}$ transitively on ${\mathbb{C}}^{\times }$, $\mathfrak{M}(1,1)$ is one point.

A monomial basis of $A_1=\mathbb{C}[i,j]\mathrm{/}(ij)$ is $1$, $i^a$ ($a\ge 1$), and $j^b$ ($b\ge 1$). Under the point action $(i,j)\mapsto (gi,j{g}^{-1})$, evaluation of these monomials scales by $1$, $g^a$, and $g^{-b}$, respectively. The nonconstant basis monomials have pairwise distinct nonzero characters $g^a$ and $g^{-b}$; their linear independence therefore implies that an invariant polynomial has only its constant term; hence $A_1^{G_1}=\mathbb{C}$. Its spectrum has one point by [F3], and the image of the stable orbit is that point, so ${\mathfrak{M}}_0(1,1)$ and $\mathcal{L}(1,1)$ are points.

The product $\mathfrak{M}(0,1)\times \mathfrak{M}(1,1)$ is a point by steps 1.1 and 1.2. It lies in $P_{0,1}$: use $V^{\mathrm{\prime }}=0$ and the stable middle datum $(V_1,I=0,J=1)$; the inclusion is injective and the quotient has one-dimensional vertex space, zero framing space and zero induced maps, hence is $S$. Thus $P_{0,1}$ is a nonempty subset of a point, so it is a point; both projections are the unique identity map.

By [F2] and [F4], each summand is one-dimensional with its point class as basis, and each point has complex dimension zero. The projections of $P_{0,1}$ are identities; hence [F1]'s point correspondence rule sends $e_0$ to $e_1$ and its reverse sends $e_1$ to $e_0$. The reverse sign is $(-1{)}^{(0-0)\mathrm{/}2}=1$; extending the maps by zero on the other summands gives the stated actions.

On $e_0$, $EF{e}_0=e_0$ and $FE{e}_0=0$; on $e_1$, $EF{e}_1=0$ and $FE{e}_1=e_1$. Thus $[E,F]$ has eigenvalues $1,-1$ on $e_0,e_1$, respectively, which are the specified $H$-weights, so $[E,F]=H$.

On $e_0$, $[H,E]e_0=H(0)-E(e_0)=0=2E{e}_0$; on $e_1$, $[H,E]e_1=H(e_0)-E(-e_1)=2e_0=2E{e}_1$. Hence $[H,E]=2E$.

On $e_1$, $[H,F]e_1=H(0)-F(-e_1)=0=-2F{e}_1$; on $e_0$, $[H,F]e_0=H(e_1)-F(e_0)=-2e_1=-2F{e}_0$. Hence $[H,F]=-2F$. In the ordered basis $(e_0,e_1)$ these actions are $E=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, $F=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$, and $H=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$; their brackets just verified are the defining brackets of the trace-zero matrix Lie algebra, so the action is its standard representation.

Define $\rho (a)(h)=ah$ for $a,h\in H$. Since multiplication is $R$-bilinear, for each fixed $a$ the map $h\mapsto ah$ is $R$-linear and $a\mapsto \rho (a)$ is $R$-linear, so $\rho :H\to {\mathrm{End}}_R(H)$ is an $R$-linear map.

For $a,b,h\in H$, $\rho (ab)(h)=(ab)h=a(bh)=(\rho (a)\circ \rho (b))(h)$ by associativity, and $\rho (1_H)(h)=1_{H}h=h$, so $\rho (1_H)={\mathrm{id}}_H$. Thus $\rho$ is a unital $R$-algebra homomorphism and hence a Hall-algebra representation by [F1].

By [F1] and [F5], scalar extension of the generic Hall algebra acts on its regular module by multiplication; extending the isomorphism [F1] from $\mathbb{Z}[t,t^{-1}]$ to $K$ identifies this action with multiplication by $p_n$ when the Hall element ${\mathrm{\Phi }}_t^{-1}(p_n)$ acts. Thus $X_n$ is the transported regular action. By [F2]–[F3], the power sums form an orthogonal basis with nonzero norms, since $z_{\lambda }\mathrm{\ne }0$ and $t^r-1\mathrm{\ne }0$ in $K$ for every part $r\ge 1$. For partitions $\alpha ,\beta$, both adjoint pairings vanish unless $\beta$ is obtained by adjoining one part $n$ to $\alpha$; in that case, if $m=m_n(\alpha )$, then $z_{\beta }=n(m+1)z_{\alpha }$ and $(X_n{p}_{\alpha },p_{\beta }{)}_G=(p_{\alpha },X_n^{†,G}{p}_{\beta }{)}_G=n(m+1)(t^n-1{)}^{-1}(p_{\alpha },p_{\alpha }{)}_G$. Thus the displayed operator is the adjoint on every pair of basis elements, hence on all of ${\mathrm{\Lambda }}_K$.

The formula in the Statement gives $P_n{p}_{\lambda }=m_n(\lambda )p_{\lambda \setminus (n)}$, the formal derivative $\mathrm{\partial }{p}_{\lambda }\mathrm{/}\mathrm{\partial }{p}_n$ by [F3], with value zero when $m_n(\lambda )=0$. Therefore the $P_n$ commute, the $X_n$ commute, and $P_{n}1=0$.

For any $m,n\ge 1$ and basis monomial $p_{\lambda }$, the polynomial product rule gives $(P_n{X}_m-X_m{P}_n)p_{\lambda }={\delta }_{nm}{p}_{\lambda }$. Since the $p_{\lambda }$ form a basis by [F3], $[P_n,X_m]={\delta }_{nm}\mathrm{id}$. Together with the commuting relations of step 1.2 these are precisely the generator brackets in [F4].

For $1\le i\le N$, send $X_i,P_i,Z$ in ${\mathfrak{h}}_N(K)$ to multiplication by $p_i$, $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{p}_i$, and $\mathrm{id}$, respectively. Step 2.1 and the commuting relations in step 1.2 preserve every defining bracket, so the assignments give a Lie algebra representation. For any nonnegative integers $m_1,\dots ,m_N$, the monomial $p_1^{m_1}\cdots p_N^{m_N}$ is ${\prod }_{i=1}^N{X}_i^{m_i}1$; hence these vectors span $K[p_1,\dots ,p_N]$ and, as $N$ varies, generate ${\mathrm{\Lambda }}_K$.

Stability and $IJ=0$ give $S=\mathrm{im}J$ of dimension $v$ and $A=JI$ with image in $S$ and $A{\mathrm{\mid }}_S=0$. Conversely such $(S,A)$ and a frame $j:V\stackrel{\sim }{\to }S$ give $J=j$ followed by inclusion and $I=j^{-1}A$. These constructions are inverse and algebraic on the graph charts [F1]; changing the frame is exactly the free $\mathrm{GL}(V)$ action. The stable locus is therefore the frame bundle over the displayed vector bundle, proving the algebraic geometric quotient claim, without an appeal to an unspecified GIT quotient. Trace identifies the dual of $\mathrm{Hom}(S,W\mathrm{/}S)$ with $\mathrm{Hom}(W\mathrm{/}S,S)$, yielding $M_v=T^{\ast }{X}_v$.

The incidence equations $A(W)\subseteq S$, $A(S)=0$ cut out a closed subscheme of $X_v\times \mathrm{End}(W)$. Projection from this product is projective, hence so is ${\mu }_v$. Its fiber at $0$ is the zero section scheme, and $A^2=0$ everywhere. This agrees with the zero fiber of the affine invariant quotient of the framed data if that quotient is also used: its invariant functions include all entries of $JI$, so the zero fiber forces $A=0$; conversely at $I=0$ the orbit closure under $g=t^{-1}{\mathrm{id}}_V$ contains the zero datum as $t\to 0$, so every invariant function has its zero-datum value. Thus no assertion identifying the entire affine invariant ring is needed.

A framed inclusion $V\hookrightarrow V^{+}$ requires $I^{+}=\xi I$ and $J=J^{+}\xi$, with the identity on $W$. In the quotient model it is exactly $S\subset T$ and a common $A$ satisfying $A(W)\subset S$, $A(T)=0$; conversely these conditions reconstruct the inclusion and its unframed one-dimensional quotient. They describe the vector bundle ${\mathcal{K}}_v$ of rank $k=v(w-v-1)$ on $Y_v$. The map $r$ is the quotient projective bundle of the tautological rank-$(v+1)$ bundle on $X_{v+1}$, with fiber ${\mathbb{P}}^v$; $q$ is the bundle of lines in $W\mathrm{/}\mathcal{S}$, with fiber ${\mathbb{P}}^{w-v-1}$. Hence $\dim Y_v=a_{v+1}+v=a_v+w-v-1$ and $\dim Z_v=\dim Y_v+k=a_v+a_{v+1}$.

A tangent vector to the flag incidence is induced by some $\xi \in \mathrm{End}(W)$, since adapted bases identify the incidence with a transitive $\mathrm{GL}(W)$ orbit. Cotangent vectors $(A_1,A_2)$ annihilate every such tangent vector exactly when $\mathrm{tr}((A_1+A_2)\xi )=0$ for all $\xi$, that is $A_2=-A_1$. Together with the two cotangent conditions this says $\mathrm{im}{A}_1\subset S$, $A_1(T)=0$. This identifies the conormal fibers, with the asserted sign in the second factor, and the same linear equations identify vector bundles on every chart. In particular $Z_v$ is smooth and closed in the product. For the first projection, $Z_v$ is a closed subvariety of $M_v\times X_{v+1}$ cut out by $S\subset T\subset \ker A$; for the second it is closed in $X_v\times M_{v+1}$ cut out by $\mathrm{im}A\subset S\subset T$. Both are projective projections. Setting either source or target $A$ to zero cuts all fiber coordinates of ${\mathcal{K}}_v$, so both inverse images of zero sections are exactly $Y_v$ with its reduced zero-section scheme structure.

An element $h\in \mathrm{GL}(W)$ acts on data by $(I,J)\mapsto (I{h}^{-1},hJ)$ and on the quotient by $(S,A)\mapsto (hS,hA{h}^{-1})$. It acts on flags and on $\mathrm{Hom}(W\mathrm{/}T,S)$ by the same transport. The frame quotient, trace pairing, incidence equations and all projections commute with these actions. This proves the equivariance assertions, including the conormal sign identification. The inherited Axiom of Choice supplies bases; the actual chart computations use only finite-dimensional choices.

The source zero-section embedding has codimension $a_v$ and normal bundle ${\mathrm{pr}}_1^{\ast }{T}^{\ast }{X}_v$. Its pullback to $Z_v$ is the zero section $Y_v\hookrightarrow \mathrm{Tot}{\mathcal{K}}_v$, with normal bundle ${\mathcal{K}}_v$ of rank $k=v(w-v-1)$. The normal map is the fiber inclusion of common $A$ into $q^{\ast }{T}^{\ast }{X}_v$. There is a canonical exact sequence $0⟶{\mathcal{K}}_v⟶q^{\ast }{T}^{\ast }{X}_v⟶{\mathrm{\Omega }}_{Y_v\mathrm{/}{X}_{v+1}}⟶0.$ Indeed tangent vectors to an $r$-fiber move $S$ inside fixed $T$; restriction of cotangent forms is the surjection dual to $T_{Y_v\mathrm{/}{X}_{v+1}}\hookrightarrow q^{\ast }T{X}_v$. Its kernel consists exactly of $A:W\mathrm{/}S\to S$ annihilating $T\mathrm{/}S$, namely $\mathrm{Hom}(W\mathrm{/}T,S)$. Thus the excess has rank $a_v-k=v$. The excess formula [F2] gives the first intersection class, of dimension $\dim Y_v-v=a_{v+1}$.

For any quotient projective bundle $p:\mathbb{P}(B)\to T$ of rank $n+1$, put $h=c_1(\mathcal{O}(1))$. Its relative Euler sequence is $0⟶{\mathrm{\Omega }}_p⟶p^{\ast }B\otimes \mathcal{O}(-1)⟶\mathcal{O}⟶0.$ The middle map evaluates the tautological quotient after tensoring by its inverse line; its kernel is the relative cotangent bundle, as one checks on a chart where one quotient coordinate is $1$. Thus [F3] gives $c_n({\mathrm{\Omega }}_p)={\sum }_{i=0}^n(-1{)}^{n-i}(n+1-i)p^{\ast }{c}_i(B)h^{n-i}$: on the splitting tower the roots are $b_j-h$, and in choosing a monomial with $i$ roots $b_j$ the remaining $n-i$ choices can be made in $n+1-i$ ways. Pushforward kills every $i>0$ term, since its hyperplane exponent is below $n$, and sends $h^n$ to $1$. Consequently $p_{\ast }(c_n({\mathrm{\Omega }}_p)\cap [\mathbb{P}(B)])=(-1{)}^n(n+1)[T]$. This also covers $n=0$.

In the other direction the same restriction to tangent vectors of a $q$-fiber gives $0⟶{\mathcal{K}}_v⟶r^{\ast }{T}^{\ast }{X}_{v+1}⟶{\mathrm{\Omega }}_{Y_v\mathrm{/}{X}_v}⟶0.$ A tangent vector moves $T$ while fixing $S$; annihilating all these variations forces the cotangent endomorphism with kernel $T$ to have image in $S$. This is again ${\mathcal{K}}_v$. Its excess rank is $a_{v+1}-k=w-v-1$, giving the second intersection class of dimension $\dim Y_v-(w-v-1)=a_v$. Neither exact sequence presumes a global splitting.

Apply step 1.2 first to $r$, the quotient projective bundle of the rank-$(v+1)$ tautological bundle on $X_{v+1}$, with $n=v$; then apply it to $q$, the bundle of lines in $W\mathrm{/}\mathcal{S}$, equivalently the quotient projective bundle of $(W\mathrm{/}\mathcal{S}{)}^{\ast }$, with $n=w-v-1$. Steps 1.1–1.2 give exactly the two asserted coefficients. All maps in the normal and Euler sequences are canonical restrictions, evaluations or differentials of equivariant projections, hence are $\mathrm{GL}(W)$-equivariant. No equivariant Chern parameter has been specialized silently: all pushforward identities just used are ordinary Chow identities.

The projections of $R_{uv}$ to either factor are projective by base change from ${\mu }_v$ or ${\mu }_u$. The intersection support defining $D\star C$ lies in $M_u{\times }_{\mathrm{End}(W)}{M}_v{\times }_{\mathrm{End}(W)}{M}_t$; forgetting the middle factor is projective by base change from ${\mu }_v$, so $p_{13}$ is proper there. The support for $C\star \alpha$ lies in $L_u\times L_v$ and projects properly to $L_v$ because $L_u$ is projective. The diagonals of the smooth $M_v$ and $M_u$ are regular of codimensions $2a_v$ and $2a_u$. Refined Gysin and pushforward therefore define the stated homomorphisms with exactly the displayed degree shifts.

For three kernels on consecutive pairs, place their external product on the six-factor product, and impose the two identifications of the repeated middle varieties by refined diagonal Gysins. The two regular embeddings use independent factors; their refined operations commute and their composite is the common double-diagonal operation by [F2]. Either parenthesization of convolution is this operation followed by forgetting the two middle factors. To move each intermediate pushforward past the remaining Gysin, use the proper base-change compatibility in [F2] on the common-endomorphism support; properness was checked in step 1.1. Functoriality [F3] then identifies the two expressions, proving associativity without a transversality assertion for the kernels.

Place $\alpha$, $C$ and $D$ on the corresponding five-factor product. The same argument imposes first the source and then the middle diagonal, or in the opposite order, with the same final projection to $L_t$. This proves the action identity. For a diagonal kernel the imposed middle equality is a regular section of a smooth projection; its refined Gysin cancels that projection's flat pullback by [F2], so the resulting operation is the identity. This proves the unit assertions as well.

Since $L_u$ and $M_u$ are smooth and $L_u\hookrightarrow M_u$ is a zero section, the diagonal intersection with $[L_u]\times C$ equals the base-changed zero-section Gysin on $C$. In local smooth coordinates this is the graph equality followed by the equations for that zero section; Gysin composition [F2] identifies it with the single regular embedding $L_u\times M_v\hookrightarrow M_u\times M_v$. The resulting classes remain supported on $L_v$ by step 1.1. Finally transport by $h\in \mathrm{GL}(W)$ preserves the common-endomorphism supports, all diagonals and all projection maps by [F1]. Proper and Gysin naturality [F2]–[F3] therefore intertwine the induced ordinary Chow actions. This proves the final assertions.