Faithful Semisimple Representations Of Reductive Groups

Hey guys! Let's dive into a fascinating question in the realm of algebraic groups: Do reductive groups have faithful geometrically semisimple representations? This is a crucial question when we're trying to understand the structure and representations of these groups, which pop up all over the place in math, from number theory to physics. We're going to unpack this, explore why it matters, and look at some of the key ideas involved. This discussion touches on some pretty deep topics in algebraic geometry and algebraic groups, so buckle up!

What are Reductive Groups and Semisimple Representations?

Before we jump into the main question, let's make sure we're all on the same page with some definitions.

Reductive Groups: Think of reductive groups as a class of algebraic groups that are "well-behaved" in terms of their representation theory. An algebraic group G over a field k is called reductive if its unipotent radical (the largest connected unipotent normal subgroup) is trivial. In simpler terms, this means that the group doesn't have any nasty unipotent subgroups messing things up. Reductive groups include many familiar groups, such as:

• General Linear Groups (GLₙ): The group of all invertible n x n matrices.
• Special Linear Groups (SLₙ): The group of n x n matrices with determinant 1.
• Orthogonal Groups (Oₙ): The group of n x n matrices that preserve a quadratic form.
• Symplectic Groups (Sp₂ₙ): The group of 2n x 2n matrices that preserve a symplectic form.

These groups are the workhorses of representation theory, and understanding their representations is fundamental.

Semisimple Representations: Now, let's talk about representations. A representation of a group G is a homomorphism (a structure-preserving map) from G into GL(V), where V is a vector space. In simpler terms, it's a way of "realizing" the group as matrices acting on a vector space. A representation is called semisimple (or completely reducible) if the vector space V can be written as a direct sum of irreducible subrepresentations. An irreducible representation is one that doesn't have any nontrivial subrepresentations – it can't be broken down further. Semisimple representations are nice because they're built from these fundamental building blocks. When we say a representation is geometrically semisimple, we mean that it remains semisimple even when we extend the field k over which our group is defined to its algebraic closure (think of extending from the rational numbers to the complex numbers).

Why do we care about geometrically semisimple representations?

The idea of geometrically semisimple representations is super important because it gives us a stable notion of semisimplicity. Sometimes, a representation might be semisimple over a field k, but when we extend to a larger field, it might break apart and become non-semisimple. Geometric semisimplicity ensures that this doesn't happen. This is particularly crucial when we're dealing with representations over fields that aren't algebraically closed, like the rational numbers or finite fields. The behavior of representations over the algebraic closure often dictates the behavior over the base field, so having this stability is a big win.

The Big Question: Faithful Geometrically Semisimple Representations

Okay, so now we're ready to tackle the main question: Do reductive groups have faithful geometrically semisimple representations? A representation is faithful if the homomorphism from G into GL(V) is injective (one-to-one). In other words, the representation "sees" the entire group – it doesn't map distinct elements to the same matrix. A faithful representation is essential because it allows us to study the group by studying its matrix representations, which are often easier to work with.

So, putting it all together, we're asking: Can we find a representation of a reductive group that:

1. Is faithful (captures the whole group).
2. Is geometrically semisimple (remains semisimple over the algebraic closure)?

This is a pretty loaded question, and the answer isn't immediately obvious. It turns out that the answer is yes, and this result has some profound implications.

Milne's Corollary 19.18 and Its Significance

Our discussion actually stems from Corollary 19.18 in Milne's Algebraic Groups, a fantastic resource for anyone diving into this subject. This corollary states that if a smooth connected algebraic group has a faithful semisimple representation that remains semisimple over the algebraic closure, then it has some special properties. Specifically, it implies something about the structure of the group and its representations.

Let's break down the significance of this corollary. It connects the existence of a faithful geometrically semisimple representation to structural properties of the group. If we can find such a representation, we gain valuable information about the group itself. This is a common theme in representation theory: representations act as a window into the inner workings of a group.

The Link to Reductivity

Milne's corollary is closely tied to the notion of reductivity. In fact, the existence of a faithful geometrically semisimple representation is a key characteristic of reductive groups. This is a powerful result because it gives us a way to characterize reductive groups in terms of their representations. We can say that a smooth connected algebraic group is reductive if and only if it admits a faithful geometrically semisimple representation. This "if and only if" statement is incredibly useful because it provides two equivalent ways of thinking about reductivity: one in terms of the group's internal structure (its unipotent radical) and another in terms of its representations.

Proof Insights and Key Ideas

While we won't go through a full, rigorous proof here (that would take us deep into the weeds of algebraic geometry!), let's touch on some of the key ideas involved in showing that reductive groups have faithful geometrically semisimple representations. The proof typically involves:

1. Constructing a Representation: The first step is to construct a representation. This often involves using the adjoint representation (the representation of the group on its Lie algebra) or other naturally occurring representations.
2. Showing Faithfulness: We need to show that the representation we've constructed is faithful. This means verifying that the map from the group to GL(V) is injective.
3. Proving Geometric Semisimplicity: This is usually the trickiest part. It involves showing that the representation remains semisimple after extending the field to the algebraic closure. This often requires using tools from algebraic geometry, such as the theory of algebraic group actions and the structure of reductive groups.

The Role of the Algebraic Closure

As we've mentioned, the algebraic closure plays a critical role in this story. Working over the algebraic closure often simplifies things because we have access to all the roots of polynomials and can use powerful tools from complex analysis (if we're working over the complex numbers). The fact that a representation remains semisimple over the algebraic closure is a strong condition that often implies good behavior over the base field as well.

Implications and Applications

So, why is this result so important? What can we do with it? The existence of faithful geometrically semisimple representations has several significant implications and applications:

Understanding Group Structure

As we've discussed, this result gives us a way to characterize reductive groups. It tells us that these groups have a rich supply of semisimple representations, which we can use to study their structure. By analyzing these representations, we can gain insights into the subgroups, conjugacy classes, and other important features of the group.

Representation Theory

This result is fundamental in representation theory. It allows us to focus on semisimple representations, which are often easier to work with than non-semisimple ones. Semisimple representations decompose into irreducible representations, which are the basic building blocks of all representations. Understanding these irreducible representations is a central goal of representation theory.

Applications in Other Areas of Mathematics

Reductive groups and their representations appear in many other areas of mathematics, including:

• Number Theory: Reductive groups are used to study automorphic forms and Galois representations, which are central objects in modern number theory.
• Algebraic Geometry: Reductive groups play a key role in the study of moduli spaces and other geometric objects.
• Physics: Reductive groups appear in gauge theory and other areas of theoretical physics.

The existence of faithful geometrically semisimple representations is a powerful tool in all of these areas. It allows us to bring the machinery of representation theory to bear on a wide range of problems.

Examples and Concrete Cases

Let's consider some concrete examples to illustrate these ideas:

1. GLₙ(k): The general linear group GLₙ(k) is a classic example of a reductive group. Its standard representation on kⁿ is faithful and geometrically semisimple. This is relatively straightforward to see: the standard representation is clearly faithful, and it remains semisimple over any field extension because it's irreducible.
2. SLₙ(k): The special linear group SLₙ(k) is another important reductive group. Its standard representation on kⁿ is also faithful and geometrically semisimple. However, things can get more interesting when we consider other representations, such as the adjoint representation.
3. Tori: A torus is an algebraic group isomorphic to a product of multiplicative groups (Gₘ)^n. Tori are reductive, and their representations are particularly well-understood. They provide a good starting point for studying the representations of more general reductive groups.

By looking at these examples, we can start to see how the general theory plays out in specific cases. We can also appreciate the importance of having concrete examples to guide our intuition.

Conclusion: A Cornerstone Result

So, to wrap things up, the question of whether reductive groups have faithful geometrically semisimple representations is a crucial one in the theory of algebraic groups. The answer is yes, and this result is a cornerstone of the theory. It connects the structure of reductive groups to their representation theory, providing us with powerful tools for studying these groups and their applications.

By having faithful geometrically semisimple representations, we gain a stable and reliable way to understand the group's behavior. This stability is essential when working over fields that aren't algebraically closed, and it allows us to apply representation-theoretic techniques in a wide range of contexts.

I hope this discussion has shed some light on this important topic. Algebraic groups and their representations are a rich and fascinating area of mathematics, and there's always more to explore. Keep asking questions, keep digging deeper, and you'll be amazed at what you discover! Cheers, guys!