AppendicesIllustrative Examples for Theorem 2

The following examples illustrate the statements and proof of Theorem 2 and help the reader follow its key steps.

1.

Consider the partition \([4^2,1^4]\). The string of even parts is at the beginning of the partition. Consequently, \(Q_1(u)\) has a leading term with coefficient 1.

$$\begin u^2 + c_4 u +c_8 = ^2 \end$$

This can be solved by setting \(a_0=1\), \(a_4 = c_4/2\). Then, we have a constraint, \(c_8 = \tfrac c_4^2\). Eliminating \(c_8\), CD and BK agree: \(S_}=S_}\)

2.

Consider the partition \([2^6]\). This is a very-even partition, so it consists of a single block of even parts. Since it starts with an even part, \(Q_1(u)\) has a leading term with coefficient 1.

$$\begin u^6+c_2 u^5+c_4 u^4+c_6 u^3 +c_8 u^2+c_ u + \tilde^2 = ^2 \end$$

We set \(a_0=1\) and solve for \(a_2=c_2/2\). The subsequent terms yield

$$\begin \begin a_2^2 +2a_0 a_4&= c_4\\ 2(a_0 a_6 +a_2 a_4)&= c_6\\ 2 a_2 a_6 + a_4^2&= c_8\\ a_6&=\pm \tilde \end \end$$

which can be successively solved for \(a_\):

$$\begin \begin a_4&=\tfrac\left( c_4-\tfracc_2^2\right) \\ a_6&= \tfracc_6-\tfrac c_2(c_4-\tfracc_2^2) \end \end$$

Plugging back in yields the constraints

$$\begin \begin c_6\mp 2\tilde&=\tfracc_2(c_4-\tfrac c_2^2)\\ c_8&=\tfracc_2^2\right) }^2\pm c_2\tilde\\ c_&=\pm \tilde(c_4-\tfracc_2^2) \end \end$$

(24)

The two solutions for \(a_6\), which yield the two choices of sign in (24), correspond to the two nilpotent orbits associated with this very-even partition. Again, CD and BK agree.

3.

Consider the partition \([4^2,2^2]\). Again, the partition is very-even, so there is a single block of even parts. However, there are two distinct even parts, so we get two perfect squares. Again, the leading part is even, so \(Q_1(u)\) has leading coefficient 1.

$$\begin \begin u^2+c_4 u +c_8&= (a_0 u+ a_4)^2\\ c_8 u^2 +c_ u + \tilde^2&= (a_4 u +a_6)^2 \end \end$$

Here, we set \(a_0=1\) and \(a_4=c_4/2\), which yields the constraint \(c_8 =\tfrac c_4^2\). Plugging that into the second equation, we have \(a_4=c_4/2\) and \(a_6= \pm \tilde\). This yields the constraint \(c_=\pm \tildec_4\). We thus have two solutions, corresponding to the two nilpotent orbits determined by this very-even partition.

4.

Consider the very-even partition \([4^2,2^4]\). We again get two perfect squares

$$ \begin u^2+c_4 u+c_8&= (u+a_4)^2\\ c_8 u^4+c_u^3+c_u^2+c_u+\tilde^2&=(a_4 u^2+a_6 u +a_8)^2 \end $$

Both BK and CD can solve for \(a_4=\tfracc_4\) and thence for \(c_8\)

From the second equation, we have two possible solutions for \(a_8\), \(a_8=\pm \tilde\), corresponding to the two choice of nilpotent orbit determined by this very-even partition. The remaining constraint equations read

$$\begin \begin c_&= c_4 a_6\\ c_&= a_6^2\pm c_4\tilde\\ c_&=\pm 2\tildea_6 \end \end$$

(25)

CD take the local Hitchin base to be the vector space spannedFootnote 15 by \(c_2,c_4,c_6,a_6,\tilde\). BK eliminate \(a_6\) from (25) and write the local Hitchin base as the singular affine variety

$$ S_}=\left\ c_^2 - 4 c_ \tilde^2 \pm 4 c_ \tilde^3&=0\\ c_ c_ \mp 2 c_ c_ \tilde + 2 c_^2 \tilde^2&=0\\ c_^2 - c_^2 c_ \pm c_^3 \tilde&=0\\ c_ c_ \mp 2 c_ \tilde&=0 \end\right\} \subset \mathbb ^7 $$

Away from the singular locus

$$ \=c_^2-4c_\tilde^2=0\} $$

the map \(\gamma : S_}\rightarrow S_}\) is 1-1. On the singular locus, it is generically 2-1 (except on the line \(\=c_=c_=0\}\) where it is 1-1), hence a normalization.

5.

Consider the partition \([5,3,2^2]\). This partition has the block of even parts at the end. Hence CD and BK disagree. We have

$$\begin c_8 u^2+ c_ u + \tilde^2 = (a_4 u +a_6)^2 \end$$

We can solve for \(a_6 =\tilde\). But the remaining constraints then read

$$\begin c_8= a_4^2 ,\qquad c_ = 2a_4 \tilde \end$$

(26)

CD say that \(a_4\) and \(\tilde\) parametrize the Hitchin image (instead of \(c_8, c_\) and \(\tilde\)). BK eliminate \(a_4\) and say this factor in the Hitchin image is parametrized by \(c_8, c_\) and \(\tilde\), with the constraint \(c_^2=4c_8\tilde^2\).

The two spaces, \(\mathcal _}\) and \(\mathcal _}\) are birational. The map \(\mathcal _}\rightarrow \mathcal _}\) is 1:1 away from the locus \(\=0,\; c_8\ne 0\}\), where it is 2:1. Again, \(\mathcal _}\) is the normalization of \(\mathcal _}\)

6.

Consider the partition \([5^2,4^2,2^4,1^2]\). We again get two perfect squares

$$ \begin c_ u^2 + c_ u + c_&= (a_5 u+ a_9)^2\\ c_ u^4 + c_ u^3+c_u^2+c_u+c_&=(a_9 u^2+a_ u +a_)^2 \end $$

In this case, equating coefficients in the polynomial equations yields:

$$ \begin c_&= a_5^2 \\ c_&= 2 a_5 a_9 \\ c_&= a_9^2 \\ c_&= 2 a_9 a_ \\ c_&= a_^2 + 2 a_9 a_ \\ c_&= 2 a_ a_ \\ c_&= a_^2 \end $$

Here BK would eliminate \(a_5\), \(a_9\), \(a_\) and \(a_ \) and write this factor in the local Hitchin base, \(S^\), as the singular affine variety which is the intersection of a quadric and seven cubics in \(\mathbb ^9\) (spanned by \(c_,c_,\dots , c_\)).

$$\begin S=\left\ c_^2 - 4 c_ c_&=0\\ c_^3 - 4c_ c_ c_ + 8c_^2 c_&=0\\ c_^2 c_ - 4 c_ c_^2 + 2 c_ c_ c_ + 16 c_^2 c_&=0\\ c_^2 c_ - 4 c_ c_ c_ + 8 c_ c_ c_&=0\\ c_ c_^2 - c_^2 c_&=0\\ c_ c_^2 - 4 c_ c_ c_ + 8 c_ c_ c_&=0\\ c_ c_^2 - 4 c_^2 c_ + 2 c_ c_ c_ + 16 c_ c_^2&=0\\ c_^3 - 4 c_ c_ c_ + 8 c_ c_^2&=0 \end \right\} \subset \mathbb ^9 \end$$

(27)

while CD would take \(S^\) to be the vector space \(\mathbb ^6\) spanned by \(a_5,a_,a_,a_,\) \(c_,c_ \). The map \(\gamma ^:S_}^\rightarrow S_}^\), is 2-1 away from the singular locus

$$ \=c_=c_=c_^2-4c_c_=0\} $$

of (27), where it is finite-1. If \(r:\,a_,a_\}\mapsto \,-a_,-a_\}\), then \(S^_}/r\) is birational to \(S^_}\).

7.

Finally, let us consider a partition with three distinct blocks of even parts, one at the beginning, one in the middle and one at the end: \([6^2,5^2,4^4,3^2,2^2]\) in \(D_\). We have

$$\begin \begin u^2+ c_6 u+ c_&= (a_0 u+a_6)^2\\ c_u^4+c_u^3+c_u^2+c_u+c_&= (a_ u^2+a_ u +a_)^2\\ c_u^2+c_u+ \tilde^2&= (a_ u +a_)^2 \end \end$$

The sign ambiguity among the a’s in the first equation can be resolved by setting \(a_0=1\). Then, we have \(2a_6=c_6\), \(a_6^2=c_\). Or

$$\begin c_ = \tfrac c_6^2. \end$$

On this, CD and BK agree.

Similarly, among \(a_,a_\), the sign ambiguity can be fixed by setting \(a_=\tilde\). Then we have \(2a_\tilde=c_\), \(a_^2=c_\).

BK would eliminate \(a_\) from these equations and write the singular hypersurface in \(\mathbb ^3\) (spanned by \(c_,c_,\tilde\))

$$\begin 4 c_ \tilde^2 = c_^2 \end$$

CD would construct the vector space, \(\mathbb ^2\), spanned by \(a_\) and \(\tilde\). The map to BK’s space is given by

$$\begin \begin c_&=a_^2\\ c_&=2a_\tilde\\ \tilde&=\tilde \end \end$$

This map is 1-1 away from the locus \(\=0,\; c_\ne 0\}\). On that locus, it is 2-1.

Finally, let us turn to the block of even parts in the middle. Both CD and BK agree on the vector space factor, spanned by \(c_,c_,c_,c_\). The constraints on the remaining c’s yield

$$\begin \begin c_&= a_^2\\ c_&= 2a_ a_\\ c_&= 2a_ a_+a_^2\\ c_&=2a_a_\\ c_&= a_^2 \end \end$$

BK eliminate \(a_,a_,a_\) and write \(S\subset \mathbb ^5\) as the singular intersection of 7 cubics:

$$\begin S=\left\ c_^3 - 4c_ c_ c_ + 8c_^2 c_&=0\\ c_^2 c_ - 4 c_ c_^2 + 2 c_ c_ c_ + 16 c_^2 c_&=0\\ c_^2 c_ - 4 c_ c_ c_ + 8 c_ c_ c_&=0\\ c_ c_^2 - c_^2 c_&=0\\ c_ c_^2 - 4 c_ c_ c_ + 8 c_ c_ c_&=0\\ c_ c_^2 - 4 c_^2 c_ + 2 c_ c_ c_ + 16 c_ c_^2&=0\\ c_^3 - 4 c_ c_ c_ + 8 c_ c_^2&=0 \end \right\} \subset \mathbb ^5 \end$$

CD take S to be the vector space \(\mathbb ^3\), spanned by \(a_,a_,a_\).

The Equivariant Fundamental Group and Lusztig’s Quotient

For the benefit of readers, we recall the formulas that describe the equivariant \(G_\)-fundamental group/component group \(A(\mathcal )\) and Lusztig’s quotient \(\overline(\mathcal )\).

\(A(\mathcal )\) in type-D (see [14, pp. 92] ) : Let a be the number of distinct odd parts in the partition label \([\mathcal ]\). Then, \(A(\mathcal ) = (\mathbb _2)^m\) where \(m = \text (a-1,0)\) if all odd parts have even multiplicity and \(m = \text (a-2,0)\) otherwise.

\(\overline(\mathcal )\) in type-D (originally defined in [42], we use the summary provided in [35] and [36]) : Let \([\mathcal ] = [\lambda _1^,\lambda _2^,\ldots ]\) and let \(\nu _i = \sum _^ a_j \). Now, identify the odd parts in \([\mathcal ]\) for which \(\nu _i\) is odd. These are sometimes called marked parts. Form a string of integers \(\mu \) using the marked parts. Let m be the number of independent, non-empty, even cardinality subsets of \(\mu \). We have that \(\overline(\mathcal ) = (\mathbb _2)^m\).

Note that the notions of a marked pair in Def 6 and that of a marked part are related but different. Every marked pair has an underlying marked part but not every marked part is part of a marked pair. This will play an important role in what follows.

From the prescription to compute Lusztig’s quotient and the definition of \(\overline_b(\mathcal )\) in Def 7, it is clear that \(\bar_b \subset \bar(\mathcal )\). More generally, we have

$$\begin \bar_b \subseteq \bar(\mathcal ) \subseteq A(\mathcal ) \end$$

Here is a table with some illustrative examples :

Hitchin Systems for 3-Punctured Spheres

Here we set out our conventions for meromorphic Hitchin systems [8,9,10] on a punctured curve C. In particular, we will give a very concrete model for type-\(D_N\) on a 3-punctured sphere.

Let \(D=\sum _i p_i\) be the divisor corresponding to the punctures. A \(D_N\) meromorphic Higgs bundle is a pair \((E,\Phi )\), composed of a \(D_\)-bundle E on C and a holomorphic section \(\Phi \) of \(ad(E) \otimes L\), where \(L=K_C(D)\), called the Higgs field. We fix \(\Phi (p_i)\in O_i\), where the \(O_i\) are a collection of nilpotent orbits in \(\mathfrak (2N)\).

The moduli space of Higgs bundles on C, \(\mathcal _\), fibers over the Hitchin base. In type-\(A_\) [11], the Hitchin base is the graded vector space \(\bigoplus _^N H^0\bigl (C,\) \(L^}(-\sum _i \chi _i^ p_i)\bigr )\). The \(\chi _i^\) are positive integers which depend on the choice of nilpotent orbits, \(O_i\). For the regular nilpotent orbit \(\chi ^=1\), whereas for the minimal nilpotent orbit \(\chi ^=k-1\).

More concretely, the sections \(\phi _k\in H^0\bigl (C,L^}(-\sum _i \chi _i^ p_i)\bigr )\) are symmetric polynomials in the Higgs field \(\Phi \)

$$ \phi _k= s_k(\Phi )=\tfrac\operatorname (\Phi ^k)+\dots $$

In type-\(D_N\), the \(\phi _k\) vanish for odd k and \(\phi _=\tilde^2\), where

$$ \tilde=\operatorname (\Phi ) $$

As sections of \(L^}\), the \(\phi _k\) vanish to order \(\chi ^_i\) at \(p_i\). In type-\(D_N\) (in contrast to type-A), the leading coefficients of the \(\phi _k\) at \(p_i\) obey local constraints that are the subject of this paper. The Hitchin base, in type-\(D_N\), is the space of solutions to those constraints.

The spectral curve \(\Sigma \rightarrow C\) is a hypersurface in the total space of L

$$\begin \Sigma =\left\ +\sum _^ w^\phi _k + \tilde^2\right\} \end$$

(28)

where w is the fiber coordinate on L. Every such \(D_N\) spectral curve admits an involution \(\iota : \Sigma \circlearrowleft \), under which w and the Seiberg-Witten differential are odd

$$ \iota ^*(w)=-w,\qquad \iota ^*(\lambda _})=-\lambda _} $$

Let \(\pi : \Sigma \rightarrow \Sigma /\iota \) be the projection; the fibers of the Hitchin map are the Prym variety \(\operatorname (\pi )\).

For the 3-punctured sphere, we can be even more concrete. Let x, y be homogeneous coordinates on \(C=\mathbb\mathbb^1\), such that the three punctures are located at \(p_1=\\), \(p_2=\\) and \(p_3=\\), respectively. The line bundleFootnote 16\(L=\mathcal (1)\rightarrow \mathbb\mathbb^1\). The \(\phi _k(x,y)\) are then homogeneous polynomials of degree 2k. Choosing three arbitrary Hitchin partitions, one for each puncture, the spectral curve takes the form of a homogeneous polynomial of degree 2N in the variables x, y, w

$$\begin \Sigma = \left\ +\sum _^\phi _(x,y)w^ +\tilde(x,y)^2 \right\} \end$$

(29)

Overall, the polynomial in (29) is homogeneous, of degree 2N in the variables (x, y, w).

For three regular nilpotents, the \(\phi _\) vanish at least linearly at each of the points and we can write

$$\begin \phi _= x y (x-y) f_(x,y),\qquad \tilde= x y (x-y) \tilde(x,y) \end$$

where the \(f_\) are homogeneous polynomials of degree \(2k-3\) and \(\tilde\) is a homogeneous polynomial of degree \(N-3\). The Seiberg-Witten differential is a meromorphic 1-form on the total space of L. In the case at hand, it has a simple expression in homogeneous coordinates

$$\begin \lambda _ = \frac \end$$

and is manifestly odd under \(\iota \). There is a second \(\mathbb ^*\) action, under which w scales with weight-1 and the coefficients in the polynomials \(f_(x,y)\) scale with weight-2k (and the coefficients in \(\tilde(x,y)\) scale with weight-N. As a consequence, \(\lambda _\) scales with weight-1. Since the coefficients in the polynomials parameterize the Hitchin base, the latter is graded by this \(\mathbb ^*\) action and the constraints are homogeneous.

The Work of Kraft-Procesi on Special Pieces

In this appendix, we will briefly review the main result of [43] and some of the tools used in the proof of this result.

To begin, we recall that nilpotent orbits in \(D_N\) can be designated by D-partitions of 2N, or alternatively by a Young diagram with columns of heights specified by the partition. The only requirement for these partitions is that even integers must appear with even multiplicity, and in cases where all integers are even, there are actually two orbits linked to that partition. Another method of representing these nilpotent orbits involves using partitions of 2N, where the corresponding Young diagrams have rows of length specified by those partitions (Hitchin). Among these orbits, there is a subset referred to as special which are those in the range of the Spaltenstein map d.Footnote 17 An alternative definition is that a partition \([\mathcal ]\) is special if the transpose \([\mathcal ]^T\) is a C-partition [14, pp. 69-70].Footnote 18 An easy way to check this is to check that there is an even number of odd parts between even parts at the beginning or ending of the partition. For example \([3^2,2^2,1^2]\) is special, while \([3,2^2,1]\) is not.

The Spaltenstein map, d takes Nahm partitions and maps them to Hitchin partitions and vice versa. However, in type D, this map is not a one-to-one correspondence. When two Nahm partitions map to the same Hitchin partition, they are considered to be part of the same special piece. In other words, a special piece is defined as the set of nilpotent orbits that map to the same special nilpotent orbit under the action of \(d^2\) (Figs. 2, 3).

Fig. 2

An example of the (a) move of Kraft-Procesi

Fig. 3

An example of the (g) move of Kraft-Procesi

There is a partial ordering on the nilpotent orbits in \(\mathfrak \), given by orbit closure. A minimal degeneration \(O\rightarrow O'\) is one where \(O'\) lies in the closure of O and there are no intervening between O and \(O'\) in the partial ordering.Footnote 19 These minimal degenerations correspond to moves, in which we rearrange the boxes in the corresponding Young tableaux. These moves were classified (a)–(g) in [43]. Moves (b)–(f) involve moving two boxes. Moves (a) and (g) are distinguished by the fact that they involve moving only one box.

Type a: For this move, the \(j^}\) column of the partition has odd height \(p_j = 2r+1\) and is followed by followed by the \((j+1)^}\) column of height \(p_ = 2r-1\). The move consists of moving one box from the \(j^}\) column to the \((j+1)^}\) column. After the move, both columns have height \(p_j=p_=2r\).

Type g: For this move, \(p_=2r+1\), \(p_=p_=\dots =p_=2r\) and \(p_=2r-1\). The move consists of moving one box from the \(j^}\) column to the \((j+2m+1)^}\) column. After the move, \(p_j=p_=\dots =p_=2r\).

These moves are what Kraft-Procesi call small degenerations. Two moves are considered independent if they can be composed. In other words, if making a move makes it impossible for us to perform a second move, those are considered to be dependent.

To understand what it means for two degenerations to be independent, consider example the \(\mathfrak \)(24) partition \(\eta = [7^2,5,3,1^2]\). A priori, we can perform three different a-moves, which we’ll differentiate by using a subindex:

$$\begin & \eta \xrightarrow \sigma _1 = [7,6^2,3,1^2] \\ & \eta \xrightarrow \sigma _2 = [7^2,4^2,1^2] \\ & \eta \xrightarrow \sigma _3 = [7^2,5,2^2,1] \end$$

Looking at \(\sigma _2\), we see that we cannot perform another \(a_1\)-move, and hence, \(a_1\) and \(a_2\) are not independent. On the other hand, moves \(a_1\) and \(a_3\) can be composed in the following way (the order does not matter)

$$\begin \eta \xrightarrow \sigma _ = [7,6^2,2^2,1] \end$$

and hence constitute independent degenerations.

Furthermore, we found it convenient to introduce the term “non-special degeneration“ (Def 10) to refer to those small degenerations which lead to non-special orbits. It is only these small degenerations that play an important role in the structure of the special piece.

Since the terminology around various degenerations can be confusing, we recall them here in one place :

Minimal degeneration (as in Kraft-Procesi [44]) : They are labeled from type (a) to (g).

Small degeneration (as in [44]) : They are minimal degenerations of type (a) or (g)

Non-special degeneration (implicit in [44], we use it explicitly) : These are small degenerations of a special Nahm orbit \(\mathcal _s\) that result in a non-special orbit \(\mathcal _\). By Spaltenstein (cited in [44]), we know that \(\mathcal _\) is in the same special piece as \(\mathcal _s\).

To see the need for the third definition above, consider the special nilpotent orbit \(\eta =[7^3,5^2,3^3,1^4]\) in \(\mathfrak (44)\) which has three independent small degenerations corresponding to the following operations :

$$\begin & \eta \xrightarrow \sigma _ = [7^2,6^2,5^2,3^3,1^4] \\ & \eta \xrightarrow \sigma _ = [7^3,5,4^2,3^2,1^4] \\ & \eta \xrightarrow \sigma _ = [7^3,5^2,3^2,2^2,1^2] \end$$

Out of this, only the third one \(\sigma _3\) is a non-special orbit and hence only \(a_3\) constitutes a non-special degeneration in the sense of Def 10.

With this terminology, we can state Theorem 6.2 of Kraft-Procesi [43] in the following way:

[Style2 Style1 Style3 Style3]Theorem 4

(Kraft-Procesi) If a special orbit \(\mathcal _s\) admits s independent non-special degenerations, then the size of the special piece containing \(\mathcal _s\) is \(2^s\).