Investigate the conditions for a projection of a Lagrangian subspace to remain Lagrangian under different subspace conditions | Step-by-Step Solution

What We're Solving:

We're exploring when the projection of a Lagrangian subspace onto another subspace preserves the Lagrangian property in symplectic vector spaces. This involves understanding the interplay between Lagrangian subspaces, coisotropic subspaces, and projections in symplectic geometry.

The Approach:

This is a beautiful problem that connects several key concepts in symplectic geometry! We'll build our understanding step by step, starting with definitions, then exploring the geometric intuition, and finally establishing the precise conditions. The key insight is that the behavior of projections depends critically on how the subspaces interact with the symplectic structure.

Step-by-Step Solution:

Step 1: Recall the fundamental definitions

• A Lagrangian subspace L satisfies: ω|_L = 0 and dim(L) = ½dim(V)
• A coisotropic subspace C satisfies: C^⊥ ⊆ C (where ⊥ is the symplectic orthogonal)
• The symplectic orthogonal of a subspace W is W^⊥ = {v ∈ V : ω(v,w) = 0 for all w ∈ W}

Step 2: Set up the projection scenario Let's say we have:

• (V,ω) a symplectic vector space
• L a Lagrangian subspace
• W a subspace we're projecting onto
• π: V → W the projection map

Step 3: Analyze what "projection remains Lagrangian" means For π(L) to be Lagrangian in W (with the induced symplectic structure), we need:

• π(L) to be isotropic in W
• π(L) to have the right dimension

Step 4: Find the key condition The crucial insight is that π(L) remains Lagrangian when W is coisotropic and L intersects W^⊥ appropriately.

Here's why: If W is coisotropic, then W/W^⊥ inherits a symplectic structure. The projection π(L) will be Lagrangian in this quotient space precisely when:

• L ∩ W^⊥ has the "expected" dimension
• The image π(L) is isotropic

Step 5: State the precise theorem The projection π(L) is Lagrangian in W (with induced symplectic structure) if and only if:

• 1. W is a coisotropic subspace
• 2. L intersects W^⊥ transversally in the appropriate sense
• 3. dim(π(L)) = ½dim(W/W^⊥)

The Answer:

A projection of a Lagrangian subspace L remains Lagrangian if and only if:

• The target subspace W is coisotropic
• The intersection L ∩ W^⊥ has dimension dim(L) + dim(W^⊥) - dim(V)/2
• This ensures π(L) becomes a Lagrangian subspace of the symplectic quotient W/W^⊥

The geometric intuition is that coisotropic subspaces are exactly those that can "carry" symplectic structure to their quotients, and Lagrangian subspaces project nicely onto these quotients when they intersect the "null space" W^⊥ in just the right way.

Memory Tip:

Think "Coisotropic Contains Compatible projections" - coisotropic subspaces are the natural targets for projections that preserve symplectic properties! The coisotropic condition ensures that the quotient space W/W^⊥ has a well-defined symplectic structure, which is essential for the notion of "Lagrangian" to make sense in the projected space.

Remember: Lagrangian subspaces are "half-dimensional and self-orthogonal" - this property can only be preserved under projection when the target space can accommodate this special structure, which is exactly what coisotropic subspaces provide!