On the Geometry of Unimodular Equivalence Classes

of Bilinear Forms

D.Ž. Djoković J. Sekiguchi K. Zhao

K an alg. closed field of characteristic 0

V fin. dim. K-vector space

dim (V) = n = 2m or 2m + 1 B the space of bilinear forms

f : V × V → K B = B^{+} ⊕ B^{−}, f = f^{+} + f^{−}

B^{+} symmetric forms
B^{−} skew-symmetric forms

GL_{n} = GL(V) SL_{n} = SL(V) {v_{1}, v_{2}, …, v_{n}} a basis of V f ∈ B A_{f} = Mat (f) = (a_{ij}) a_{ij} = f(v_{i}, v_{j}) B is a GL_{n}-module (a, f) → a · f, a ∈ GL_{n}, f ∈ B (a · f)(x, y) = f(a^{−1}(x), a^{−1}(y)), ∀ x, y ∈ V B^{+} , B^{−} are submodules GL_{n}-orbits in B are equivalence classes of forms SL_{n} · f = {a · f : a ∈ SL_{n}} unimodular or SL_{n}-orbits K[B] the K-algebra of regular (polynomial) functions on B K[B]^{SL_{n}} the subalgebra of SL_{n}-invariant regular functions P (f, t) := det (A_{f^{+}} − tA_{f^{−}}) f ∈ B, t an indeterminate P (f, t) = P_{0}(f) + P_{1}(f)t^{2} + ⋯ + P_{m}(f)t^{2m} If n = 2m is even then P_{m}(f) = Pf(A_{f^{−}})^{2}. Pf is the Pfaffian Theorem (Adamovich & Golovina, 1977) K[B]^{SL_{n}} is generated by

P_{0}, P_{1}, …, P_{m}  if n = 2m + 1
P_{0}, P_{1}, …, P_{m−1}, Pf if n = 2m.

In both cases these generators are algebraically independent.

The embedding

K[B] ⟵ K[B]^{SL_{n}}

correponds to the projection map

π : B ⟶ B//SL_{n} ≅ K^{m+1} .

B//SL_{n} is the categorical quotient. The morphism π is onto.

π(f) = (P_{0}(f), P_{1}(f), …, P_{m}(f)),  n = 2m + 1
= (P_{0}(f), P_{1}(f), …, Pf(f^{−})),  n = 2m

The fibers π^{−1}(π(f)) ⊂ B are closed subvarieties of B.

N := π^{−1}(π(0)) is the null-cone.

The fibers are equi-dimensional varieties and all of them have dimension

n^{2} − m − 1.

If f ∈ N , f is a null-form.

Example. n = 2

A_{f} = (\begin{matrix}a & b \\ c & d \end{matrix})

P (f, t) = det ((\begin{matrix}a & \frac{b+c}{2} \\ \frac{b+c}{2} & d \end{matrix} ) − t (\begin{matrix}0 & \frac{b−c}{2} \\ \frac{c−b}{2} & 0 \end{matrix}))

P_{0}(f) = ad − \frac{1}{4}(b + c)^{2} Pf(f^{−}) = \frac{1}{2}(b − c) N is defined by the equations:

b = c, ad = b^{2}.

N is the union of two SL_{2}-orbits:

1) {0} and

2) symmetric matrices of rank 1.

Questions about the SL_{n}-action on B:

1) Is N the union of only finitely many SL_{n}-orbits?

2) Characterize the closed SL_{n}-orbits.

3) Given f ∈ B, determine the unique closed SL_{n}-orbit contained in π^{−1}(π(f)).

4) For which f ∈ B one has π^{−1}(π(f)) = SL_{n} · f?

5) Is there an SL_{n}-orbit which is open in N?

6) Is N irreducible?

7) Construct a Weierstrass section.

Proposition 2.3 If n ≥ 3, N contains infinitely many SL_{n}-orbits.

Pf. α ∈ K, α^{2} ≠ −1, f_{α} ∈ B

Mat (f_{α}) = (\begin{matrix}A_{α} & 0 \\ 0 & 0 \end{matrix}) , A_{α} = (\begin{matrix}α & 1 \\ −1 & α \end{matrix})

f_{α} ≅ f_{β} ⇒ XA_{α}X′ = A_{β} for some X ∈ GL_{2}

A_{α}^{−} = A_{β}^{−} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ⇒ X ∈ SL_{2}

β^{2} + 1 = |A_{β}| = |A_{α}| = α^{2} + 1 ⇒ α^{2} = β^{2} Theorem 7.3

The orbit SL_{n} · f, f ≠ 0, is closed in B iff, with respect to a suitable basis, the matrix of f is the direct sum of

(1) 1-by-1 blocks (λ), λ ≠ 0

and

(2) 2-by-2 blocks (\begin{matrix}0 & μ \\ ν & 0 \end{matrix}) , μ ≠ ν.

D(f) := discriminant of

|tA_{f^{+}} − A_{f^{−}}|

(a polynomial in t of degree n) B_{gen} := {f ∈ B : |A_{f^{+}} | ≠ 0 ≠ D(f)} Theorem 7.5 π^{−1}(π(f)) = SL_{n} · f ⟺ f ∈ B_{gen} Proof of ⇐ May assume A_{f^{+}} = λI_{n}, λ ∈ K^{∗}. Define u^{−} ∈ End (V) by

f^{−}(x, y) = f^{+} (u^{−}(x), y), ∀ x, y ∈ V. Mat (u^{−}) = −λ^{−1}A_{f^{−}} D(f) ≠ 0 ⇒ u^{−} has n distinct eigenvalues u^{−} ∈ 𝔰o_{n} := 𝔰o(f^{+} ) regular ss element May assume

A_{f−} = ⊕_{i=1}^{m} [\begin{matrix}0 & μ_{i} \\ −μ_{i} & 0 \end{matrix}]

⊕[0] if n = 2m + 1

μ_{i}^{2} nonzero & distinct Hence

(SO_{n})_{f^{−}} = T = a max. torus of SO_{n} and

(SL_{n})_{f} = (SO_{n})_{f^{−}} = T.

Observe dim (SL_{n} · f) = n^{2} − 1 − m = dim π^{−1}(π(f)).

How to find the unique closed orbit, say SL_{n}·f_{c}, contained in the given fiber π^{−1}(π(f))? If f ∈ N , then f_{c} = 0. Assume f ∉ N . Find a decomposition

V = V_{1} ⊥ V_{2} ⊥ ⋯ ⊥ V_{k}
f = f_{1} ⊥ f_{2} ⊥ ⋯ ⊥ f_{k}

such that (V_{i}, f_{i}) is indecomposable. We can take

f_{c} = (f_{1})_{c} ⊥ (f_{2})_{c} ⊥ ⋯ ⊥ (f_{k})_{c}.

Indecomposable cases

f  f_{c}

(1) [\begin{matrix}0 & I_{m} \\ J_{m}(λ) & 0 \end{matrix}] , λ ≠ 0, (−1)^{m+1}  [\begin{matrix}0 & I_{m} \\ λI_{m} & 0 \end{matrix}]

(2) [ \begin{matrix}& & 1 \\ & −1 & −1 \\ 1 & 1 \end{matrix} ]  [ \begin{matrix}& & 1 \\ & −1 \\ 1 \end{matrix} ]

[ \begin{matrix}& & & −1 \\ & & 1 & 1 \\ & −1 & −1 \\ 1 & 1 \end{matrix} ]  [ \begin{matrix}& & & −1 \\ & & 1 \\ & −1 \\ 1 \end{matrix} ]

Γ_{n}, n ≥ 1

(3) J_{n}(0), n = 2m
[\begin{matrix}0 \\ 1 & 0 \\ & 1 & 0 \\ & & 1 & 0 \end{matrix}]  [ \begin{matrix}0 & 0 \\ 1 & 0 \\ & & 0 & 0 \\ & & 1 & 0 \end{matrix} ]

f = \begin{aligned}& & & 1 \\ & & & & 1 \\ & & & & & 1 \\ λ \\ 1 & λ \\ & 1 & λ \end{aligned} , X = \begin{aligned}1 \\ & t^{−1} \\ & & t^{−2} \\ & & & 1 \\ & & & & t \\ & & & & & t^{2} \end{aligned}

XfX′ = \begin{aligned}& & & 1 \\ & & & & 1 \\ & & & & & 1 \\ λ \\ t & λ \\ & t & λ \end{aligned} , f_{c} = \begin{aligned}& & & 1 \\ & & & & 1 \\ & & & & & 1 \\ λ \\ & λ \\ & & λ \end{aligned}

Γ_{3} = \begin{matrix}0 & 0 & 1 \\ 0 & −1 & −1 \\ 1 & 1 & 0 \end{matrix} , X = \begin{aligned}t^{−1} \\ & 1 \\ & & t \end{aligned}

XΓ_{3}X′ = \begin{matrix}0 & 0 & 1 \\ 0 & −1 & −t \\ 1 & t & 0 \end{matrix} , Γ_{3}^{+} = \begin{aligned}& & 1 \\ & −1 \\ 1 \end{aligned}

f = Γ_{n}  ,  f_{c} = f^{(−)^{n+1}} .

The matrix of the form f_{0}

[ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]  [ \begin{matrix}& & 1 \\ & 0 & 1 \\ −1 & 0 \end{matrix} ]

[ \begin{matrix}& & & 1 \\ & & 0 & 1 \\ & 0 & 1 \\ −1 & 0 \end{matrix} ]  [ \begin{matrix}& & & 0 & 1 \\ & & & 1 & 1 \\ & & 0 & 1 & 0 \\ 0 & −1 & 0 \\ −1 & 0 & 0 \end{matrix} ]

[ \begin{matrix}& & & & 0 & 1 \\ & & & & 1 & 1 \\ & & & & 1 & 0 \\ & & & 1 \\ 0 & −1 & 0 \\ −1 & 0 & 0 \end{matrix} ]  [ \begin{matrix}& & & & 0 & 0 & 1 \\ & & & & 0 & 1 & 1 \\ & & & & 1 & 1 & 0 \\ & & & 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & −1 & 0 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ]

The orbit SL_{n} · f_{0} ⊂ N which is open in N .

(1) n = 2m + 1 J_{n}(0) ∈ SL_{n} · f_{0}

(2) n = 2m J_{n}(0) ∉ N J_{n−1}(0) ⊕ [1] ∈ SL_{n} · f_{0} Weierstrass section of π : B → B//SL_{n} ≅ K^{m+1}

[ \begin{matrix}ξ & η \\ 0 & 1 \end{matrix} ]  (ξ − η^{2}/4, η/2)

[ \begin{matrix}ξ & η & 1 \\ 0 & 0 & 1 \\ −1 & 0 & 0 \end{matrix} ]  (−ξ/4, ξ/4 − η)

[ \begin{matrix}ξ & η & ζ & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ]  \begin{matrix}(−ξ/4 + ζ^{2}/16, \\ ξ/4 − η − ζ^{2}/8, ζ/4) \end{matrix}

[ \begin{matrix}ξ & η & ζ & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 \end{matrix} ]  \begin{matrix}(ξ/16, −ξ/8 + η/4 + ζ/4, \\ ξ/16 − η/4 + 3ζ/4) \end{matrix}

Main Theorem

(1) N is irreducible.

(2) Each fiber π^{−1}(π(f)) is irreducible.

(3) Any nonzero fiber of π contains only finitely many SL_{n}-orbits. (The number of these orbits is bounded.)

Proof based on the paper: V.L. Popov, The cone of Hilbert nullforms, Proc. Steklov Inst. Math. 241 (2003), 177–194.

Intersection of an orthogonal and a symplectic group

Let f ∈ B with both f^{+} and f^{−} nondegenerate. Then n = 2m.

(GL_{n})_{f} = (GL_{n})_{f^{+}} ∩ (GL_{n})_{f^{−}}
= O(f^{+} ) ∩ Sp(f^{−})
= SO(f^{+} ) ∩ Sp(f^{−})

Proposition 9.1. For G = (GL_{n})_{f}, m ≤ dim G ≤ m^{2}.

(1) dim G = m ⇒ G is a direct product of a torus and an abelian unipotent group.

(2) dim G = m^{2} ⇒ G ≅ GL_{m}.

Linear operators E Bilinear forms

B

u ∈ E semisimple

iff SL_{n} · u is closed f ∈ B semisimple means that SL_{n} · f is closed

u ∈ E nilpotent f ∈ N is a null-form

Jordan decomposition u = u_{s} + u_{n} u_{s} semisimple u_{n} nilpotent u_{s}u_{n} = u_{n}u_{s} Decomposition is

unique f = f_{s} + f_{n} f_{s} semisimple f_{n} null-form Decomposition is not unique!

Problem. Is there a simple condition that f_{s} and f_{n} should satisfy to make the decomposition unique?

Jordan Decomposition of Bilinear Forms

D.Ž. Djoković K. Zhao f ∈ B Z(f) := (GL_{n})_{f} centralizer of f T_{f} := sℓ_{n} · f tangent space to O_{f} := SL_{n} at f. g ∈ B semisimple

→ Z(g) reductive

B = T_{g} ⨁ N N = Z(g) – invariant subspace N_{g} ⊂ N the null–cone for the action of

Z(g) ∩ SL_{n} = (SL_{n})_{g} on N.

Definition of Jordan decomposition f ∈ B , f = g + h (J1) g ∈ C_{f}

C_{f} := the unique closed SL_{n}–orbit in \overline{O_{f}}.

(J2) h ∈ N_{g} for some choice of N, B = T_{g} ⨁ N (N is Z(g)– invariant)

(J3) Z(f) = Z(g) ∩ Z(h).

K = an algebraically closed field char K = 0 P^{1}K = K ∪ {∞}, projective line ι : P^{1}K → P^{1}K ι(λ) = λ^{−1}. Z_{2} = ⟨ι⟩ Π = P^{1}K/Z_{2} Definition: μ : P^{1}K → P^{1}K is a ι – map if

1) ιμ = μι

2) μ(1) = −1 , μ(−1) = 1. To each ι–map, μ, we associate a Jordan decomposition f = g_{μ} + h_{μ} Primary Decomposition For f ∈ B\N there is a unique decomposition

(V, f) = ⊥_{π ∈ Π} (V^{π}, f^{π}).

such that V^{{0,∞}} is totally degenerate and otherwise (V^{π} , f^{π}) is nondegenerate and π is the set of eigenvalues of the asymmetry σ^{π} of f^{π}.

Theorem

(a) f ∈ B\N , f = g + h a Jordan decomposition, g semisimple. Then f and g have the same primary decompositons

V = ⊥_{π ∈ Π} V^{π} .

f^{π} = g^{π} + h^{π} is a Jordan decomposition ∀ π ∈ Π.

(b) Converse holds.

Primary Case π ∈ Π, π = {λ, λ^{−1}}, λ ≠ λ^{−1}. f ∉ N , f = f^{π}. Define φ := \frac{f−λf′}{1−λ}. Then

φ ∉ N , φ is {0, ∞}-primary .

L(W) = {v ∈ V : φ(v, W) = 0} R(W) = {v ∈ V : φ(W, v) = 0} L(V) ⊂ L^{3}(V) ⊂ L^{5}(V) ⊂ ⋯ ⊂ L_{∞}(φ) R(V) ⊂ R^{3}(V) ⊂ R^{5}(V) ⊂ ⋯ ⊂ R_{∞}(φ) V_{λ}(f) := L_{∞}(φ), V_{1/λ}(f) := R_{∞}(φ) V = V_{1/λ}(f) ⊕ V_{λ}(f) f = g_{μ} + h_{μ} μ-Jordan decomposition μ = a ι-map

g_{μ} = [\begin{matrix}0 & \frac{f′−μ(λ)f}{λ−μ(λ)} \\ λ · \frac{f−μ(λ)f′}{λ−μ(λ)} & 0 \end{matrix} ]

h_{μ} = [\begin{matrix}0 & \frac{λf−f′}{λ−μ(λ)} \\ μ(λ) · \frac{λf′−f}{λ−μ(λ)} & 0 \end{matrix} ]

g_{μ}, h_{μ} are independent of the choice of λ ∈ π. If π ≠ {0, ∞}, then V_{λ}(f) is the generalized λ-eigenspace of the asymmetry of f.

Three special ι-maps:

1) Jordan decomposition of symmetric type

μ_{+} (λ) = 1  if λ ≠ 1
= −1 if λ = 1

2) Jordan decomposition of skew-symmetric

type

μ_{−}(λ) = −1 if λ ≠ −1
= 1  if λ = −1

3) Jordan decomposition of harmonic type

μ_{0}(λ) = λ^{−1} if λ ≠ ±1
= −1  if λ = 1
= 1  if λ = −1