# Orbits and Schubert varieties in the affine Grassmannian

*Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Public domain (CC0).*

A relative position records how far a lattice has moved. Its orbit is smooth, but its compactification can acquire singularities. We will compute the orbit directions, prove the closure order and component classification, and describe the first singular Schubert surface by an equation. This gives concrete geometry for the intersection complexes used in the later lessons.

Throughout, \(k\) is algebraically closed, \(O=k[[t]]\), \(F=k((t))\), and \(G\) is connected reductive. Fix \(T\subset B\subset G\), with positive roots those of \(B\). Write \(K=G(O)\), \(\rho=\frac12\sum_{\alpha>0}\alpha\), and \(t^\lambda=\lambda(t)\). Dominance means \(\langle\alpha,\lambda\rangle\geq0\) for every positive root. Our Schubert varieties have the **reduced** structure on orbit closures. This qualification matters in positive characteristic and for torus factors.

The construction of lattice families is in Loop groups and the affine Grassmannian. The torsor interpretation and relative positions are in Beauville–Laszlo gluing and the moduli interpretation. The earlier programme lesson Roots and reductive groups of rank one, Theorem 7.1, proves the root subgroups and the rank-one homomorphisms \(SL_2\to G\), in every characteristic. Root data, Weyl chambers and the Bruhat decomposition, §5, proves their ordered products, big-cell coordinates and polynomial commutator identities; Theorem 6.2 proves ordinary Bruhat decomposition.

Theorem 7.1 of Loop groups and the affine Grassmannian constructs the full ind-projective Grassmannian for every connected reductive group, in every characteristic. Its closed finite-type lattice stages supply the ambient schemes used for orbit closures and specialization arguments in this lesson.

## 1. Elementary divisors classify lattices

For \(GL_n\), write a coweight as
\[
\lambda=(\lambda_1,\ldots,\lambda_n),\qquad
\lambda_1\geq\cdots\geq\lambda_n.
\]
The point \(t^\lambda\) is the lattice \(\bigoplus_i Ot^{\lambda_i}e_i\).

**Theorem 1.1 (Cartan decomposition for \(GL_n\)).** Every lattice in \(F^n\) is \(GL_n(O)\)-equivalent to exactly one such lattice. Consequently
\[
GL_n(F)=\coprod_{\lambda_1\geq\cdots\geq\lambda_n}
GL_n(O)t^\lambda GL_n(O).
\]

*Proof.* Give a lattice an \(O\)-basis. Its columns form an invertible Laurent matrix \(A\). Multiply by a power of \(t\) to make every entry integral. Choose a nonzero entry of smallest valuation, move it to the upper left, and multiply its row by a unit to make it a power \(t^m\). Every other entry is divisible by \(t^m\). Integral elementary column operations clear the remainder of its row; row operations then clear its column. The remaining block still has entries divisible by \(t^m\). Repeat on that block. We obtain a diagonal matrix with nondecreasing valuations. Removing the initial common power and reversing the order proves existence.

For uniqueness, the ideal generated by the \(r\)-by-\(r\) minors is preserved by invertible integral row and column operations. On a diagonal matrix its valuation is the sum of the \(r\) smallest exponents. These sums recover each exponent by subtraction. Thus the sorted tuple is unique. Left operations change the ambient frame; right operations change the lattice basis. This is precisely the asserted double-coset classification. \(\square\)

For a general reductive group the Cartan statement is
\[
G(F)=\coprod_{\lambda\in X_*(T)^+}Kt^\lambda K. \tag{1.2}
\]
The general split-group Cartan argument is the affine-Iwahori argument in Spherical Hecke algebras as functions on the affine Grassmannian: it proves the affine double-coset normal form from the earlier root-coordinate and ordinary Bruhat proofs, then takes finite-Weyl double cosets. It works over every field, in every characteristic. Theorem 1.1 gives a separate direct proof for lattices. Zhu's freely accessible introduction, §2.1, equation (2.1.1), is a further-reading reference for this statement.

## 2. The stabilizer and the finite orbit directions

Let \(\operatorname{Gr}^{\lambda}=Kt^\lambda\) and
\[
H_\lambda=K\cap t^\lambda Kt^{-\lambda}.
\]
Conjugation on a root group satisfies
\[
t^\lambda u_\alpha(z)t^{-\lambda}
 =u_\alpha(t^{\langle\alpha,\lambda\rangle}z).
\]
Hence its integral parameter in \(H_\lambda\) must lie in
\[
t^{\max(0,\langle\alpha,\lambda\rangle)}O. \tag{2.1}
\]
Torus arcs lie in the stabilizer. Write
\[
P_\lambda^-=\langle T,\ U_\alpha:
 \langle\alpha,\lambda\rangle\leq0\rangle.
\]
Reduction modulo \(t\) sends \(H_\lambda\) onto \(P_\lambda^-\).

Here is why these parameter computations control the whole stabilizer. For \(h\in K\), the condition \(t^{-\lambda}ht^\lambda\in K\) says that conjugation by the one-parameter subgroup \(\lambda\) has no negative powers. On the special fibre it is exactly the parabolic limit condition. A lift of a point of \(P_\lambda^-\) can be made using its constant Levi and negative root factors. After dividing by this lift we are in the first congruence group \(K_1\). Every element of \(K_1\) lies in the big-cell neighbourhood of the identity, and has a unique product of negative root coordinates, torus coordinates, and positive root coordinates. Substituting \(t^{-\lambda}ht^\lambda\) imposes (2.1) on these coordinates. Uniqueness of the big-cell factorization prevents cancellation between root coordinates. This argument works over every \(k\)-algebra and at each finite jet level.

Choose \(M\) larger than every positive \(\langle\alpha,\lambda\rangle\). Then \(K_M\subset H_\lambda\), and
\[
\operatorname{Gr}^{\lambda}
 = (K/K_M)/(H_\lambda/K_M). \tag{2.2}
\]
Both groups in (2.2) are smooth of finite type. The stabilizer has a connected parabolic quotient and a connected kernel with successive additive root-coordinate quotients. It is therefore connected. The quotient is a smooth variety.

**Theorem 2.3.** The stabilizer \(H_\lambda\) is connected, and
\[
\dim\operatorname{Gr}^{\lambda}
 =\sum_{\alpha>0}\langle\alpha,\lambda\rangle
 =\langle2\rho,\lambda\rangle.
\]

*Proof.* The preceding finite-jet description proves connectedness and smoothness. Taking Lie algebras gives the orbit tangent space
\[
\mathfrak g(O)/
 \bigl(\mathfrak g(O)\cap
        \operatorname{Ad}(t^\lambda)\mathfrak g(O)\bigr)
 \cong
 \bigoplus_{\alpha>0}
 \mathfrak g_\alpha\otimes_k
       O/t^{\langle\alpha,\lambda\rangle}O.
\]
There is no torus contribution; a root of nonpositive pairing contributes zero. Counting the coefficients in these truncated series gives the dimension formula. \(\square\)

Evaluation gives a projection
\[
\operatorname{Gr}^{\lambda}\longrightarrow G/P_\lambda^-.
\]
Its fibre is \(K_1/(K_1\cap H_\lambda)\), an affine space of dimension
\[
\sum_{\alpha>0,\ \langle\alpha,\lambda\rangle>0}
       (\langle\alpha,\lambda\rangle-1).
\]
To see the affine-space assertion, order the root coefficients by increasing \(t\)-degree and root height. Divide successively by allowed stabilizer coefficients. The remaining coefficients are unrestricted affine coordinates; commutators involve only later coefficients in this order. Local sections of \(G\to G/P_\lambda^-\) over the usual big-cell charts identify the projection with that affine-space fibre over each chart. Its transition functions can contain nonlinear terms. Thus this is an affine-space fibration; the calculation alone does not give a vector bundle.

If \(\lambda\) is minuscule, every positive pairing is \(0\) or \(1\). There are no surviving congruence coefficients, so evaluation identifies the orbit with \(G/P_\lambda^-\). The latter is projective. Since the affine Grassmannian is separated, this orbit is closed.

## 3. The closure order for lattices

For two decreasing \(n\)-tuples with equal sum, write \(\mu\leq\lambda\) when
\[
\sum_{i=1}^r\mu_i\leq\sum_{i=1}^r\lambda_i
 \quad(1\leq r<n).
\]
Equivalently, \(\lambda-\mu\) is a nonnegative integral sum of the simple coroots \(e_i-e_{i+1}\). Equality of total sums is part of this definition.

**Theorem 3.1.** For \(GL_n\),
\[
\overline{\operatorname{Gr}^{\lambda}}
 =\coprod_{\mu\leq\lambda}
       \operatorname{Gr}^{\mu}.
\]

*Proof of the upper bound.* Put \(s_r(\lambda)=\lambda_{n-r+1}+\cdots+\lambda_n\). Every point of \(\operatorname{Gr}^{\lambda}\) satisfies
\[
\bigwedge^r\Lambda\subset
 t^{s_r(\lambda)}\bigwedge^r O^n.
 \tag{3.2}
\]
In any bounded lattice Grassmannian, these are closed incidence conditions: the map from the universal exterior-power lattice to the indicated quotient must vanish. They persist under specialization. The determinant component is fixed as well; applying the determinant to a lattice gives a map to the torus Grassmannian, whose reduced components are the individual integers, as proved in the previous lesson.

For a diagonal lattice of type \(\mu\), (3.2) says \(s_r(\mu)\geq s_r(\lambda)\). Together with equal sums, these are exactly the displayed dominance inequalities. This proves that every orbit in the closure has type \(\mu\leq\lambda\).

*Proof of the lower bound.* Suppose \(a-b\geq2\). In two selected coordinates use the family
\[
A(s)=
\begin{pmatrix}
t^{a-1}&0\\
s t^b&t^{b+1}
\end{pmatrix},\qquad s\in\mathbb A^1.
\tag{3.3}
\]
The determinant is \(t^{a+b}\), so the columns define a lattice family over \(k[s]\). For \(s\ne0\), the smallest entry valuation is \(b\), and the elementary-divisor type is \((a,b)\). At zero it is \((a-1,b+1)\). Keeping the other diagonal coordinates fixed gives a degeneration that transfers one unit from a larger exponent to a smaller one.

These transfers reach every \(\mu\leq\lambda\). Indeed, if \(\lambda\ne\mu\), choose the first index \(i\) where they differ. Then \(\lambda_i>\mu_i\). Let \(j>i\) be the first index where the running difference
\[
D_j=\sum_{h=1}^j(\lambda_h-\mu_h)
\]
returns to zero. Such an index exists because \(D_n=0\), and \(D_h\geq1\) for \(i\leq h<j\). We have \(\lambda_j<\mu_j\). Since \(\mu_i\geq\mu_j\), it follows that \(\lambda_i-\lambda_j\geq2\). Transfer one unit from \(i\) to \(j\), then sort the tuple again. Before sorting all running differences remain nonnegative; sorting can only increase the sums of the largest \(r\) entries. The sorted tuple therefore still dominates \(\mu\). Balancing two entries cannot increase any largest-entry sum relative to \(\lambda\), so it is dominated by \(\lambda\).

The sum of absolute differences from the sorted tuple \(\mu\) falls by at least two. Before sorting it falls by two; sorting minimizes that sum against a decreasing tuple, as follows by exchanging any inverted pair. The process terminates at \(\mu\). Each step is the degeneration (3.3). Orbit closures are \(K\)-stable, so containment of the diagonal point gives containment of its entire orbit. This proves the lower bound. \(\square\)

For \(GL_2\), this reads
\[
\overline{\operatorname{Gr}^{(a,b)}}
 =\coprod_{0\leq r\leq\lfloor(a-b)/2\rfloor}
       \operatorname{Gr}^{(a-r,b+r)}.
\tag{3.4}
\]
For general reductive \(G\), the corresponding order is
\[
\mu\leq\lambda
\quad\Longleftrightarrow\quad
\lambda-\mu\in\sum_{\alpha\ {\rm simple}}
                     \mathbb Z_{\geq0}\alpha^\vee.
\]

**Theorem 3.5 (the closure order).** For every dominant coweight \(\lambda\),
\[
\overline{\operatorname{Gr}^{\lambda}}
 =\coprod_{\substack{\mu\in X_*(T)^+\\\mu\leq\lambda}}
       \operatorname{Gr}^{\mu}.
\tag{3.5}
\]

*The rank-one degeneration.* It is useful to allow intermediate coweights that are not dominant. Let \(\nu\in X_*(T)\), let \(\alpha\) be a positive root, and suppose
\[
m=\langle\alpha,\nu\rangle\geq1.
\]
Consider the polynomial family of loops
\[
g(s)=u_{-\alpha}(s t^{1-m})t^{\nu-\alpha^\vee},
\qquad s\in\mathbb A^1.
\tag{3.6}
\]
At zero its point is \(t^{\nu-\alpha^\vee}\). For every nonzero \(s\), its point is in the \(K\)-orbit of \(t^\nu\). Here is the complete matrix check. In the rank-one calculation one can write the diagonal exponents as \((a,b)\), where \(a-b=m\). The relevant matrix and two determinant-one integral matrices are
\[
A(s)=
\begin{pmatrix}t^{a-1}&0\\s t^b&t^{b+1}\end{pmatrix},
\quad
L(s)=\begin{pmatrix}0&s^{-1}\\-s&t^{m-1}\end{pmatrix},
\quad
R(s)=\begin{pmatrix}1&-s^{-1}t\\0&1\end{pmatrix}.
\]
Their product is
\[
L(s)A(s)R(s)=\operatorname{diag}(t^b,t^a).
\tag{3.7}
\]
All entries of \(L(s)\) and \(R(s)\) are integral because \(m\geq1\). More explicitly, if \(n_\alpha\) is the image of \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\), then
\[
L(s)=\alpha^\vee(s^{-1})n_\alpha^{-1}
                 u_\alpha(-s^{-1}t^{m-1}),
\qquad R(s)=u_\alpha(-s^{-1}t).
\]
These formulas make sense in \(G\). Use the rank-one identity with the upper parameter \(x=-s^{-1}t^{m-1}\) and lower parameter \(-x^{-1}=s t^{1-m}\); their product has zero upper-left entry and factors through the interchange matrix. Commuting the remaining upper factor through \(t^{\nu-\alpha^\vee}\) multiplies it by \(t^{m-2}\), so it cancels the factor in \(R(s)\). The remaining torus factor is exactly \(t^{\nu-m\alpha^\vee}=t^{s_\alpha\nu}\), proving (3.7) in \(G\) as well. This checks the identity even when \(\nu\) does not factor through the rank-one torus. Both reflection representatives and the two multiplying factors belong to \(K\). Thus (3.6) is the required degeneration in every characteristic. For \(m=1\), its two torus points are Weyl conjugate and already in the same orbit; this step will be needed in the chain argument.

*The chain argument.* Suppose \(\mu\) is dominant and
\[
\nu-\mu=\sum_i d_i\alpha_i^\vee,
\qquad d_i\in\mathbb Z_{\geq0},
\quad \nu\ne\mu.
\]
Average a positive definite inner product over the finite Weyl group. On the semisimple real space, \((\alpha_i^\vee,x)\) is a positive multiple of \(\langle\alpha_i,x\rangle\): the reflection fixes the corresponding root hyperplane and reverses its coroot normal. Put \(\delta=\nu-\mu\). Since \(\mu\) is dominant,
\[
(\delta,\nu)=(\delta,\mu)+\|\delta\|^2>0.
\]
As \(\delta\) is a nonnegative sum of simple coroots, some index with \(d_i>0\) therefore has \(\langle\alpha_i,\nu\rangle>0\). Its pairing is an integer, hence at least one. Apply (3.6) and replace \(\nu\) by \(\nu-\alpha_i^\vee\). The remaining difference from \(\mu\) is still a nonnegative simple-coroot sum, and \(\sum_i d_i\) has decreased by one. Repetition reaches \(\mu\). The intermediate coweights may be nondominant; their torus points and their \(K\)-orbits are nevertheless defined. Starting at \(\lambda\), transitivity of closure containment proves that every \(\operatorname{Gr}^\mu\) with \(\mu\leq\lambda\) occurs in its closure.

For example, in \(SL_3\) the chain from \((1,0,-1)\) to zero can pass through \((0,1,-1)\). The first simple-root pairing is one, so that step is a Weyl conjugacy; the next pairing is two and gives the degeneration to zero. Requiring every intermediate coweight to remain dominant would exclude this elementary chain.

*The upper bound.* We first build the representations that detect all dominance inequalities, without importing a highest-weight theorem in characteristic zero. For each simple root \(\alpha_i\), let \(U_i\) be the unipotent radical of the standard maximal parabolic omitting that root. In \(\bigwedge^{\dim U_i}\mathfrak g\), the top wedge of \(\mathfrak u_i\) is a vector \(v_i\) of weight
\[
\chi_i=\sum_{\substack{\alpha>0\\
             \text{the coefficient of }\alpha_i\text{ is positive}}}\alpha.
\]
The group \(U^+\) fixes this wedge: it normalizes \(\mathfrak u_i\), acts triangularly with diagonal entries one in ordered root coordinates, and hence has determinant one. The reflections in all the other simple roots permute the displayed root set, so \(\langle\chi_i,\alpha_j^\vee\rangle=0\) for \(j\ne i\). The remaining pairing is positive. Indeed, if it were nonpositive, the squared norm of this nonzero positive-root sum would be nonpositive, by pairing its simple-root expansion with \(\chi_i\). Consequently \(\chi_i=c_i\omega_i\), where \(c_i>0\) and \(\omega_i\) is the corresponding rational fundamental weight.

Let \(V_i\) be the finite-dimensional subrepresentation generated by \(v_i\). Its weights are \(\chi_i\) minus nonnegative sums of positive roots. To check this directly, the big cell \(U^-TU^+\) is dense in \(G\), so its action on \(v_i\) spans the same space as the action of \(G\). The positive factor fixes \(v_i\), and an ordered negative-root factor has polynomial coefficients of weights \(\chi_i-n\alpha\), \(n\geq0\), by torus equivariance. Iteration proves the claim. Thus the largest weight pairing on \(V_i\) for any dominant \(\tau\) is exactly \(\langle\chi_i,\tau\rangle\).

For a point represented by \(g\), put \(\Lambda_i(g)=\rho_i(g)(V_i\otimes O)\). Every point of the orbit of \(\lambda\) satisfies
\[
t^{\langle\chi_i,\lambda\rangle}(V_i\otimes O)
       \subset\Lambda_i(g).
\tag{3.8}
\]
This is equivalent to requiring every coefficient of
\(\rho_i(g^{-1})\) below degree \(-\langle\chi_i,\lambda\rangle\) to vanish. In a bounded lattice Grassmannian it is a closed incidence condition, and therefore holds on the orbit closure. At \(t^\mu\), it gives
\[
\langle\chi_i,\mu\rangle\leq
\langle\chi_i,\lambda\rangle
\quad\text{for every }i.
\tag{3.9}
\]
The specialization invariant proved in §5 also gives
\(\lambda-\mu\in\mathbb Z\Phi^\vee\). Write this difference in the simple-coroot basis. Since \(\chi_i=c_i\omega_i\) with \(c_i>0\), (3.9) says precisely that each integral coefficient is nonnegative. Hence \(\mu\leq\lambda\).

Cartan decomposition says that every point belongs to one of the displayed orbits. Their closures are \(K\)-stable, so the lower and upper bounds prove (3.5). We take reduced structures throughout; equality of their closed sets in a bounded finite-type stage gives equality of their reduced closed subschemes. Theorem 7.1 of the preceding loop-group lesson makes the ambient bounded stages projective. These reduced closed finite-type Schubert varieties are therefore projective for every connected reductive group. \(\square\)

## 4. The first singular surface

Consider \(GL_2\) and \(\lambda=(1,-1)\). The closure has its two orbits: the dimension-two orbit and the base point \(\Lambda_0=O^2\). Every lattice in it lies between \(tO^2\) and \(t^{-1}O^2\). In
\[
V=t^{-1}O^2/tO^2
\]
use the ordered basis
\[
a=t^{-1}e_1,\quad b=t^{-1}e_2,\quad
c=e_1,\quad d=e_2.
\]
Multiplication by \(t\) is \(\phi(a)=c,\ \phi(b)=d,\ \phi(c)=\phi(d)=0\). A lattice corresponds to a \(\phi\)-stable two-plane \(W\subset V\).

Apart from \(W_0=\ker\phi=\langle c,d\rangle\), every such plane has
\[
\ell=W\cap\ker\phi=\phi(W)
\]
a line. Identify \(\ker\phi\) with \(k^2\). Then
\[
W=\langle uc+vd,\ ua+vb+z\rangle,
\qquad z\in k^2/\ell,\quad \ell=\langle(u,v)\rangle.
\]
The choice of lift defines a homomorphism \(\ell\to k^2/\ell\). Thus the open orbit is the total space of
\[
\operatorname{Hom}(\mathcal O(-1),\mathcal O(1))
 =\mathcal O_{\mathbb P^1}(2).
\]

Let \(p_{ij}\) be Plücker coordinates. These planes satisfy
\[
p_{12}=0,\qquad p_{14}=p_{23},\qquad
p_{13}p_{24}=p_{14}^2.
\]
The first two are linear equations, and the last is the remaining Plücker relation. Conversely, a nonvertex point satisfying them has \((p_{13},p_{14},p_{24})\) on the Veronese conic, and the preceding formula recovers its plane; the vertex is \(W_0\). Both the cone and the Schubert closure are reduced, so this identification of their closed points in the same projective Grassmannian identifies their reduced closed subschemes. Therefore
\[
\overline{\operatorname{Gr}^{(1,-1)}}\cong
\{[x:y:z:w]\in\mathbb P^3:xz=y^2\}. \tag{4.1}
\]
The vertex is \([0:0:0:1]\). In the affine chart \(w=1\), its tangent space has dimension three, whereas the surface has dimension two. It is singular. Away from the vertex one of \(x,z\) is nonzero, and the equation eliminates one coordinate; the surface is smooth there. This argument remains valid in characteristic two.

For \(SL_2\), determinant trivialization identifies its reduced quasi-minuscule Schubert closure with this same surface. The diagonal type is \(\alpha^\vee=(1,-1)\). A determinant-zero \(GL_2\) lattice over a field admits a basis with determinant \(1\), so the orbit and its base point agree with the \(SL_2\) points. Their reduced closures in the bounded Grassmannian agree. Multiplication of lattices by \(t\) also identifies (4.1) with the \(GL_2\) closure of type \((2,0)\).

The cone has a useful resolution. Compactify the line bundle \(\mathcal O(2)\) by its infinity section; this is the Hirzebruch surface \(\mathbb F_2\). The infinity section has normal bundle \(\mathcal O(-2)\), and the map to (4.1) contracts it to the vertex. The open orbit is the complementary \(\mathcal O(2)\). In characteristic zero with characteristic-zero sheaf coefficients this is the familiar \(A_1\) surface singularity.

## 5. Components and small groups

The determinant degree labels the components of \(\operatorname{Gr}_{GL_n}\). To prove that one degree gives only one component, start with any tuple of that degree. Repeated balancing transfers, using (3.3), reach the unique decreasing tuple whose entries differ by at most one. The corresponding orbit is a connected ordinary Grassmannian. Every orbit of that degree has closure meeting it, and each orbit is connected. Their union is therefore connected. Different degrees are separated by the determinant map to the discrete reduced torus Grassmannian. We obtain
\[
\pi_0(\operatorname{Gr}_{GL_n})=\mathbb Z.
\]

For \(SL_2\), the dominant coweights are \(m\alpha^\vee=(m,-m)\), \(m\geq0\), and the dimensions are \(2m\). Smith reduction can be performed with determinant-one ambient and lattice basis operations: adjust the determinants of the two change-of-basis matrices by diagonal units, which commute with the diagonal normal form. The closure relation is \(0\leq r\leq m\), and all orbit closures meet the base point. Thus its Grassmannian is connected.

For \(PGL_2\), coweights are indexed by \(n\geq0\), represented by \(\operatorname{diag}(t^n,1)\) modulo scalar matrices. A point is a rank-two lattice up to homothety. Every \(PGL_2(F)\) point lifts to \(GL_2(F)\), since the obstruction is a line bundle over the field \(F\). Smith reduction therefore proves the classification. Scalar homothety changes determinant degree by an even integer, giving the invariant \(n\bmod2\). The odd component has closed minimal orbit \(\mathbb P^1\), of type \(1\), and the even one has the base point. Formula (3.4) gives the orbit closure types \(n,n-2,\ldots\).

For the topological claim over all characteristic, one can use the reduced homothety-lattice functor: locally lift a homothety class to a lattice; changing the lift shifts its determinant degree by two. Thus determinant degree modulo two is locally constant and descends. This separates the two components, while the degenerations just given connect every orbit to its minimal one. Nilpotent structure of \(PGL_2\) in characteristic two does not change connected components.

For general \(G\), the component theorem is
\[
\pi_0(\operatorname{Gr}_G)
 =\pi_1(G):=X_*(T)/\mathbb Z\Phi^\vee.
\tag{5.1}
\]
The symbol \(\pi_1(G)\) here denotes the algebraic root-datum fundamental group. We now prove both the locally constant class invariant and connectedness of each of its fibres. The proof includes characteristics dividing the order of the finite part of \(\pi_1(G)\).

*A torus-kernel extension.* The earlier lesson Pinnings and the classification of split reductive groups, Theorem 10.1 and its following derived-group section, gives a presentation
\[
G=(G_{\mathrm{sc}}\times C)/H,
\]
where \(C\) is a split torus, \(G_{\mathrm{sc}}\) is semisimple simply connected, and \(H\) is finite central diagonalizable. Embed \(H\) into a split torus \(S\): a finite list of generators of its character group gives a surjection from a free abelian group, hence this closed immersion. Divide
\(G_{\mathrm{sc}}\times C\times S\) by the graph of this embedding of \(H\). The earlier affine central-quotient proof constructs the resulting reductive group \(\widetilde G\), and projection gives
\[
1\longrightarrow S\longrightarrow\widetilde G
\longrightarrow G\longrightarrow1.
\tag{5.2}
\]
Its derived subgroup is the embedded \(G_{\mathrm{sc}}\): the graph intersects this factor trivially, and the earlier proof shows that this factor is perfect, while its quotient is a torus. Write \(A=\widetilde G/G_{\mathrm{sc}}\). For the inverse-image maximal torus \(\widetilde T\), the coroots are a basis of the cocharacter lattice of its simply connected semisimple subtorus. The exact sequences of split tori therefore give
\[
X_*(A)=X_*(\widetilde T)/\mathbb Z\Phi^\vee,
\qquad
X_*(A)/\operatorname{im}X_*(S)
   =X_*(T)/\mathbb Z\Phi^\vee.
\tag{5.3}
\]
Cocharacter surjectivity in these sequences follows by dualizing their character sequences: the character group of the kernel torus is free, so the sequences split as abelian groups. No finite central isogeny is assumed to be étale.

*The invariant on field points.* An element \(g\in G(F)\) lifts to \(\widetilde g\in\widetilde G(F)\), since its lifting torsor is a split-torus torsor over a field and every line bundle there is trivial. Map the lift to \(A(F)\), take the valuations of its torus coordinates, and reduce the resulting element of \(X_*(A)\) modulo \(\operatorname{im}X_*(S)\). By (5.3) this defines
\[
\kappa(g)\in X_*(T)/\mathbb Z\Phi^\vee.
\]
Changing the lift multiplies it by an \(S(F)\)-point, so the class is independent of the lift. Multiplication of lifts shows that \(\kappa\) is a homomorphism. Integral points lift integrally because line bundles over the local ring \(O\) are free; hence \(\kappa(K)=0\). It therefore descends to the Grassmannian, and \(\kappa(t^\lambda)=[\lambda]\).

*Preservation under specialization.* Let a family be defined over the complete discrete valuation ring \(R=k[[s]]\). Its \(G\)-torsor on \(\operatorname{Spec}R[[t]]\) is trivial: the ring is complete local, its residue-field torsor has a point because \(k\) is algebraically closed, and smoothness lifts that point successively modulo powers of \((s,t)\). Thus the family is represented by \(g\in G(R[[t]][1/t])\).

This element lifts through (5.2). Its lifting torsor is a product of line-bundle torsors, and
\[
\operatorname{Pic}(k[[s,t]][1/t])=0.
\]
This is exactly Proposition 5.2 of the earlier programme lesson Regular local rings: it proves that a rank-one projective over a localization of a regular local ring is stably free by localizing a finite free resolution, then free by its top exterior power. Its §6 proves that \(k[[s,t]]\) is regular local. These are the needed algebraic statements, with actual preceding proofs.

The image of the lift in \(A\) has invertible coordinates in \(R[[t]][1/t]\). Each such unit is uniquely \(t^n u\), where \(u\in R[[t]]^\times\): if it and its inverse have bounded denominators, their numerators multiply to a power of the prime element \(t\); successive cancellation of that prime gives the assertion. The factor \(u\) is a unit on both the generic and special fibres. Thus each integer valuation is the same on both fibres, and so is \(\kappa\).

This also tests boundary points of orbit closures. A closed point in the closure of a locally closed finite-type orbit is reached by a curve with generic point in that orbit. For completeness, in an affine neighbourhood cut its irreducible closure by successive general hyperplanes through the point. At every step choose a hyperplane that contains no positive-dimensional component of the boundary; the infinite field permits avoidance of those finitely many linear conditions. After \(d-1\) cuts the closure has a curve through the point and the boundary there is zero-dimensional, so a curve component through the point meets the orbit. Normalize that curve and complete a local ring above the point to obtain \(k[[s]]\). Finite normalization and this discrete valuation ring and completion calculation are proved in Completing a smooth affine curve and extending its group action, Lemmas 1.2 and 2.1 and the paragraph following Lemma 2.1. Hence every orbit-closure specialization preserves the coroot class. This is the specialization statement used in the upper bound of Theorem 3.5; it does not use that closure theorem.

The class is locally constant. Indeed, a bounded finite-type stage meets only finitely many Cartan orbits. To see finiteness, use a faithful representation and bounds \(t^N V_O\subset\Lambda\subset t^{-N}V_O\); every weight pairing of its diagonal Cartan point then lies between \(-N\) and \(N\). The weights of a faithful torus representation span the rational character space, so only finitely many integral coweights satisfy these bounds. The orbits are locally closed finite-jet homogeneous spaces. The closure of each orbit has just been shown to retain its class; with finitely many orbits, the union belonging to one class and its complement are both closed. This proves local constancy at each bounded stage and therefore on the ind-scheme. Reduction changes no topological points, so it changes no connected components.

*Connectedness of a class.* The group \(K\) is connected: its evaluation quotient is connected \(G\), and its congruence kernels have successive additive Lie-algebra quotients, as in the earlier loop-group construction. Its finite-dimensional orbits are therefore connected. For every root \(\alpha\), the rank-one homomorphism gives a morphism
\[
\operatorname{Gr}_{SL_2}\longrightarrow\operatorname{Gr}_G.
\]
The \(SL_2\) quasi-minuscule closure from §4 is connected and contains the base point and \(t^{\alpha^\vee}\). Translating its image by \(t^\eta\) connects \(t^\eta\) with \(t^{\eta+\alpha^\vee}\), for every coweight \(\eta\). Reversing the same connection permits subtraction. Hence any two torus points whose difference is in the coroot lattice lie in the same connected component. Cartan decomposition puts every point in a connected orbit containing a torus point. Each fibre of \(\kappa\) is consequently connected, and every coroot class occurs. Local constancy separates distinct classes, proving (5.1). \(\square\)

Dimensions have constant parity on a coroot class. Indeed,
\[
\langle2\rho,\alpha_i^\vee\rangle=2
\]
for each simple coroot. This explains why orbit dimensions within a connected component have the same parity. It does not assert that all orbits of the Grassmannian have even dimension: \(\mathbb P^1\subset\operatorname{Gr}_{GL_2}\) has dimension one.

## 6. Exercises with solutions

**Exercise 6.1 (easy).** Compute the \(GL_2\) orbit dimensions of \((1,0)\), \((2,0)\), and \((1,-1)\).

*Solution.* There is one positive root, with pairing \(a-b\). The dimensions are respectively \(1,2,2\). The first orbit is \(\mathbb P^1\). The latter two have isomorphic closures by central translation, and each closure is the surface (4.1). \(\square\)

**Exercise 6.2 (easy).** Identify the \(GL_n\) orbit of \((1,0,\ldots,0)\).

*Solution.* Such a lattice satisfies
\[
tO^n\subset\Lambda\subset O^n,\qquad
\dim_k(O^n/\Lambda)=1.
\]
It is the inverse image of a hyperplane in \(k^n\). Thus it is the dual projective space \(\mathbb P((k^n)^*)\), isomorphic to \(\mathbb P^{n-1}\). This description works for families using a locally free rank-one quotient, so it identifies schemes. The orbit is closed and has dimension \(n-1\). \(\square\)

**Exercise 6.3 (medium).** Determine the equation and singular locus of the \(SL_2\) quasi-minuscule closure.

*Solution.* Use the \(\phi\)-stable two-plane description in §4. The equations \(p_{12}=0\) and \(p_{14}=p_{23}\) leave the Plücker equation \(p_{13}p_{24}=p_{14}^2\). This is the cone \(xz=y^2\) in \(\mathbb P^3\). Every nonvertex point is smooth, because either \(x\) or \(z\) permits elimination. At the vertex the local equation has no linear term, so the Zariski tangent dimension is three instead of two. The singular locus of the reduced surface is exactly its vertex, in every characteristic. The two orbits are its smooth complement and that vertex. \(\square\)

**Exercise 6.4 (medium).** Compute the tangent space of the affine Grassmannian at \(t^\lambda\), and distinguish it from the orbit tangent space.

*Solution.* A first-order deformation is represented locally by
\[
(1+\epsilon X)t^\lambda,\qquad
X\in\mathfrak g((t)),\quad \epsilon^2=0.
\]
Changing its right integral frame identifies \(X\) modulo
\(\operatorname{Ad}(t^\lambda)\mathfrak g[[t]]\). Hence, for the full quotient functor,
\[
T_{t^\lambda}\operatorname{Gr}_G
 =\mathfrak g((t))/
       \operatorname{Ad}(t^\lambda)\mathfrak g[[t]].
\]
This is generally infinite-dimensional; even a torus has such infinitesimal directions. Restricting \(X\) to \(\mathfrak g[[t]]\) gives the finite-dimensional orbit tangent quotient in Theorem 2.3. It is incorrect to equate the two. When passing to the reduced ind-scheme, nilpotent tangent directions can disappear; the displayed full-functor tangent space need not be the tangent space of its reduction. \(\square\)

**Exercise 6.5 (hard).** Prove the \(GL_2\) closure relation without using a general reductive-group theorem.

*Solution.* Fix \(d=a+b\). The closed bounds \(t^aO^2\subset\Lambda\subset t^bO^2\) and determinant degree \(d\) restrict an elementary-divisor tuple \((u,v)\) to
\[
a\geq u\geq v\geq b,\qquad u+v=d.
\]
These are precisely \((u,v)=(a-r,b+r)\) with \(0\leq r\leq\lfloor(a-b)/2\rfloor\). They give the necessary bound. Family (3.3) supplies the first next orbit, and its central translates with \(a,b\) replaced by \(a-1,b+1\) supply every subsequent one. Stability under \(GL_2(O)\) gives the whole orbit at each step. Thus (3.4) is the exact closure, with its reduced structure. \(\square\)

## References

Further reading is freely available in Xinwen Zhu, [*An introduction to affine Grassmannians and the geometric Satake equivalence*](https://arxiv.org/abs/1603.05593v2), §§1.3 and 2.1; Edward Frenkel, [*Lectures on the Langlands program and conformal field theory*](https://arxiv.org/abs/hep-th/0512172v1), §5.5, for the \(PGL_2\) example; and Laurent Fargues and Peter Scholze, [*Geometrization of the local Langlands correspondence*](https://arxiv.org/abs/2102.13459), VI.2, for the related mixed-characteristic geometry. The arguments above use ordinary power series.