Symplectic Linear Group Action

• Symplectic linear group action is the canonical representation of Sp(2n, ℝ) on 2n-dimensional vector spaces, preserving the nondegenerate skew-symmetric bilinear form.
• It underlies orbit classification by generating invariants such as Pfaffians and rational/differential invariants, critical in invariant and representation theory.
• The action extends to affine, conformal, and contact settings, offering robust tools for normal form characterization and applications in algebraic K-theory.

A symplectic linear group action is the canonical linear action of the symplectic group $\operatorname{Sp}(2n, \mathbb{R})$ (or its variants over other base fields) on symplectic vector spaces and their associated geometric and algebraic objects. This action underlies much of the structural theory of symplectic geometry, invariant theory, representation theory, and algebraic geometry related to alternating forms, and provides a source of canonical invariants and orbit classification for submanifolds, functions, module objects, and algebraic varieties.

1. The Symplectic Group and Its Linear Action

Let $V \simeq \mathbb{R}^{2n}$ be a $2n$-dimensional real vector space equipped with the standard nondegenerate skew-symmetric bilinear form

$\omega = \sum_{i=1}^n dx^i \wedge dy^i,$

where $(x^1,\ldots,x^n, y^1,\ldots, y^n)$ serves as linear coordinates. In matrix terms, setting $z = (x, y)^T$ and $$\Omega = \begin{pmatrix} 0 & I_n \ -I_n & 0 \end{pmatrix}$$, the form is $\omega(z, w) = z^T \Omega w$. The real symplectic group is then

$$\operatorname{Sp}(2n, \mathbb{R}) = \{ g \in \operatorname{GL}(2n, \mathbb{R}) \mid g^T \Omega g = \Omega \},$$

acting linearly via $z \mapsto g z$, preserving $\omega$ and therefore the symplectic structure on $V$ (Pelayo, 2016).

This construction generalizes straightforwardly to arbitrary fields, to higher tensor powers (for "joint invariants"), and to the context of modules over commutative rings with $2 \in R^\times$ (Chattopadhyay et al., 2011, Pandey et al., 2023).

2. Algebraic and Differential Invariants

2.1 Polynomial and Rational Invariants

For the diagonal action of $\operatorname{Sp}(2n,\mathbb{R})$ on $k$-tuples $V^k$, the first fundamental theorem (FFT) of symplectic invariant theory asserts that the algebra of invariants is generated by the pairwise symplectic pairings

$$I_{ij} = \omega(v_i, v_j), \quad 1 \leq i < j \leq k,$$

where $v_i \in V$ (Andreassen et al., 2020). The minimal relations (second fundamental theorem, SFT) are given by vanishing of all Pfaffians of size $2n+2$. That is, for any $2n+2$ indices, the Pfaffian of the corresponding principal submatrix of $(I_{ij})$ must vanish, with precise forms in low dimensions being the classical Plücker relation for $n=1$.

The field of rational invariants has transcendence degree $2nk - n(2n+1)$ for $k \geq 2n$ (Andreassen et al., 2020).

2.2 Differential Invariants via the Lie–Tresse Theorem

When $\operatorname{Sp}(2n, \mathbb{R})$ acts on jets of submanifolds or graphs of functions, the Lie–Tresse theorem supplies a finite generating set of differential invariants and invariant derivations. For $u = u(x, y)$ on $V$, the algebra $\mathcal{A}$ of differential invariants is generated by

• $I_0 = u$ (order 0)
• $I_1 = \sum_{i=1}^{n} (x^i u_{x^i} + y^i u_{y^i})$ (order 1)
• Quadratic contractions of the horizontal Hessian $Q_2$ as $I_{2,a}$, $I_{2,b}$, $I_{2,c}$ (order 2)
• $2n$ invariant derivations $\nabla_j$ up to structure equations and syzygies

For example, for $n=1$, the algebra closes with a finite number of differential syzygies among $I_0$, $I_{2,c}$, and the derivations $\nabla_1, \nabla_2$ (Jensen et al., 2020).

In higher codimension, the pattern is: curves in $\mathbb{R}^{2n}$ yield one second-order invariant and $n$ derivations; hypersurfaces yield $2n-1$ second-order invariants and $2n-1$ derivations, and so forth.

3. Orbit Structure and Classification

3.1 Linear and Symplectic Equivalence

The symplectic linear group action partitions ambient space (e.g., $V^k$) into orbits classified by the values of the joint invariants $I_{ij}$, subject to Pfaffian relations (Andreassen et al., 2020). Two generic $k$-tuples are in the same $\operatorname{Sp}(2n,\mathbb{R})$-orbit if and only if corresponding $I_{ij}$ invariants agree and all Pfaffian syzygies are satisfied.

In the module-theoretic context, orbits of the elementary symplectic transvection group on unimodular vectors coincide setwise with those of the elementary linear transvection group, reflecting a precise equivalence between linear and symplectic orbits once a hyperbolic summand is added (Chattopadhyay et al., 2011). This identification is critical for $K_1$-theoretic stability results.

3.2 Conjugacy of Symplectic Forms

When $\operatorname{Sp}(2n, \mathbb{R})$ acts by conjugation on the space of nondegenerate skew-symmetric matrices, the algebraic invariants are the Pfaffian and the coefficients $\sigma_k(A)$ of $\operatorname{Pf}(tJ + A)$ for $k=0,\ldots,n-1$. These $n$ invariants are algebraically independent and suffice to classify orbits in dimension four; the classification in higher dimensions is governed by the same set of invariants (Shi et al., 2022).

In the $n=2$ case, the pair $(p,q) = (\mathrm{Pf}(A), s(A))$ completely determines the orbit, with the topology of orbits described via a discriminant $\Delta = q^2/4 - p$.

4. Invariant Theory of Symplectic Group Actions on Polynomial Rings and Nullcones

The natural action of $\operatorname{Sp}_{2n}(K)$ on $m$-copies of its defining representation endows the coordinate ring $S = K[y_{i,j}]$ with an invariant subring generated by the quadratic forms

$$X_{ij} = \sum_{a,b=1}^{2n} y_{a,i} \Omega_{ab} y_{b,j},$$

subject to vanishing of all principal Pfaffians of size $2n+2$, yielding a generic Pfaffian algebra structure (Pandey et al., 2023). The Hilbert nullcone is defined by the vanishing of all nonconstant invariants, and its scheme-theoretic properties (strong F-regularity, rational singularities, properties of the divisor class group and Gorenstein locus) are completely described.

Table: Structure of the Symplectic Invariant Algebra on $m$ Copies of $V$

Generators	Relations	Singularities
Degree-2 $X_{ij}$ (from $\Omega$)	All Pfaffians of size $2n+2$ vanish	F-regular/rational
		Divisor class: $\mathbb{Z}$ if $m \ge n+1$

5. Geometric and Representation-Theoretic Consequences

The transitive $\operatorname{Sp}(n)$-action on quaternionic projective space $\mathbb{H}\mathbb{P}^{n-1}$ and flag manifolds $\operatorname{Sp}(n)/\operatorname{Sp}(1)^n$ encodes interaction degrees of freedom, with coset representatives parameterized by off-diagonal blocks, and the flag spaces serving as models for highest-weight representations via Borel–Weil theory. The Maurer–Cartan form decomposes curvature contributions, and the associated invariants reflect representation-theoretic labels (weights, roots, etc.) (Eichinger, 2011).

For analytic symplectic actions of semisimple Lie algebras, local linearization in Darboux coordinates is always possible; equivalently, any such analytic action with a fixed point is linearizable (but this generally fails in the smooth non-compact case) (Miranda, 2015).

6. Extensions: Conformal, Affine, and Contact Symplectic Actions

Modifying the group to the conformal symplectic group $$C\operatorname{Sp}(2n,\mathbb{R}) = \operatorname{Sp}(2n, \mathbb{R}) \times \mathbb{R}^+$$ brings an additional scaling generator, and invariants/derivations compatible with this (weight-zero) reduction yield generators for conformal-invariant algebras (Jensen et al., 2020, Andreassen et al., 2020). For the affine symplectic group $$\operatorname{ASp}(2n) = \operatorname{Sp}(2n) \ltimes \mathbb{R}^{2n}$$, invariants are constructed via reduction to the linear case after translation.

Contactifications lead to rational relative invariants under the extended action, and discretization of joint invariants recovers the differential invariants of submanifold geometry.

7. Characterization of Normal Forms and Orbit Moduli

Normal forms for linear symplectic transformations are governed by Williamson's theorem: every element can be brought into block-diagonal form with blocks of elliptic, hyperbolic, or focus-focus type, and possibly nilpotent Jordan blocks. Co-adjoint orbits, via the momentum map, realize symplectic quotients corresponding to the orbit structure (Pelayo, 2016).

In the context of symplectic modules over rings, normal forms and conjugacy classes inform the stabilization of $K_1$ and the structure and generation of the elementary symplectic group, with important implications for algebraic $K$-theory (Chattopadhyay et al., 2011).

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The symplectic linear group action organizes the invariant theory of symplectic structures, underlies the classification of forms and submanifolds, and connects geometric, algebraic, and representation-theoretic domains through its orbit structure, generating invariants, and their syzygies (Jensen et al., 2020, Andreassen et al., 2020, Shi et al., 2022, Pandey et al., 2023, Pelayo, 2016).