Character Dependence Of Weil Representation Unveiled

Introduction to Weil Representation

In the intricate landscape of modern mathematics, Weil representations stand as a pivotal bridge connecting representation theory, number theory, and physics. These representations, named after the eminent mathematician André Weil, offer profound insights into the structure of symplectic and orthogonal groups over local fields. This article delves deep into the character dependence of Weil representations, particularly focusing on the interplay between symplectic and orthogonal spaces. To fully grasp the nuances of this topic, it's essential to first understand the foundational concepts. The Weil representation, a cornerstone in the study of automorphic forms and representation theory, serves as a crucial tool for analyzing the symmetries and structures within these mathematical frameworks. Its construction involves intricate relationships between symplectic and orthogonal groups, providing a rich tapestry of mathematical objects to explore. This article will dissect the character dependence of Weil representations, elucidating the connections between the representation's character and the underlying algebraic structures. Understanding the character dependence of Weil representations is paramount for researchers and enthusiasts alike, as it unveils the deep-seated connections between representation theory, number theory, and physics. These representations, which act as a bridge between symplectic and orthogonal groups, offer valuable insights into automorphic forms and related mathematical structures. By exploring how the characters of Weil representations are influenced by the underlying algebraic structures, we gain a more profound appreciation for the elegance and complexity of these mathematical objects. This exploration not only enriches our theoretical understanding but also paves the way for practical applications in diverse fields, highlighting the enduring significance of Weil representations in contemporary mathematical research.

Foundational Concepts

To embark on our exploration, let's first lay the groundwork by defining some essential terms. Let $F$ be a p-adic local field. A p-adic local field is a topological field that is locally compact, non-discrete, and totally disconnected. These fields play a crucial role in number theory and are fundamental to understanding the arithmetic of algebraic number fields. A symplectic space, often denoted as $W$, is a vector space equipped with a non-degenerate, alternating bilinear form. In simpler terms, it's a space where we can define a notion of area-preserving transformations. Mathematically, if we have a symplectic space $W$, it is a vector space over a field $F$ together with a bilinear form $\langle , \rangle : W \times W \rightarrow F$ that satisfies the following properties:

1. Alternating: $\langle v, v \rangle = 0$ for all $v \in W$.
2. Non-degenerate: If $\langle v, w \rangle = 0$ for all $w \in W$, then $v = 0$.

A 2-dimensional symplectic space is the simplest non-trivial example, and it forms the basis for many constructions in representation theory. An orthogonal space, denoted as $V$, on the other hand, is a vector space equipped with a non-degenerate symmetric bilinear form. This means we have a notion of orthogonality or perpendicularity within the space. Formally, if $V$ is an orthogonal space, it is a vector space over a field $F$ together with a bilinear form $( , ) : V \times V \rightarrow F$ that satisfies:

1. Symmetric: $(v, w) = (w, v)$ for all $v, w \in V$.
2. Non-degenerate: If $(v, w) = 0$ for all $w \in V$, then $v = 0$.

In this context, we consider $V$ to be a 1-dimensional orthogonal space, which simplifies the analysis while still capturing essential features of the Weil representation. The metaplectic group, denoted as $\Mp$, is a central extension of the symplectic group. It arises naturally in the study of quadratic forms and plays a vital role in the construction of the Weil representation. The metaplectic group can be seen as a “double cover” of the symplectic group, providing a more nuanced understanding of its representation theory. To further clarify the landscape, let’s delve into the concept of the Weil representation itself. In essence, the Weil representation is a representation of the metaplectic group $\Mp$ on a suitable Hilbert space. It provides a concrete realization of the abstract symmetries encoded in the metaplectic group, linking it to more tangible mathematical structures. The Weil representation, also known as the oscillator representation, emerges from the interplay between symplectic and orthogonal spaces. It offers a powerful lens through which we can examine the symmetries and structures inherent in these spaces. Understanding the Weil representation requires a grasp of the underlying symplectic and orthogonal groups, as well as the metaplectic group, which serves as a central extension of the symplectic group. This representation maps elements of the metaplectic group to operators on a Hilbert space, thereby translating abstract algebraic structures into concrete linear transformations. This mapping allows us to study the properties of the metaplectic group through the more familiar framework of linear algebra and functional analysis. The construction of the Weil representation involves intricate details, including the choice of additive characters and the use of Schrödinger representations. These technical aspects are crucial for ensuring that the representation behaves as expected and maintains its fundamental properties. The Weil representation is not just a theoretical construct; it has practical applications in various areas of mathematics and physics, including number theory, automorphic forms, and quantum mechanics.

Setting the Stage: Symplectic and Orthogonal Spaces

Let's consider $F$ as a p-adic local field, a fundamental concept in number theory. These fields possess a unique structure that makes them indispensable in the study of algebraic number theory and representation theory. A crucial aspect of $p$-adic fields is their non-Archimedean nature, which leads to interesting arithmetic properties and distinguishes them from the more familiar real and complex numbers. In the context of Weil representations, $p$-adic fields provide a rich landscape for exploring the interplay between algebraic structures and representation-theoretic phenomena. Now, let $W$ be a 2-dimensional symplectic space over $F$. This space is characterized by a non-degenerate, alternating bilinear form. In simpler terms, a symplectic space is a vector space equipped with a way to measure oriented areas. The 2-dimensional case is particularly significant because it is the simplest non-trivial example, yet it encapsulates many of the key features of higher-dimensional symplectic spaces. The symplectic structure allows us to define transformations that preserve these oriented areas, leading to the concept of the symplectic group. The symplectic group, denoted as $Sp(W)$, consists of all linear transformations on $W$ that preserve the symplectic form. This group plays a crucial role in both classical and quantum mechanics, as well as in the theory of automorphic forms. Understanding the structure and representations of the symplectic group is essential for grasping the deeper aspects of Weil representations. Turning our attention to orthogonal spaces, let $V$ be a 1-dimensional orthogonal space over $F$. Unlike symplectic spaces, orthogonal spaces are characterized by a symmetric bilinear form, which allows us to measure lengths and angles. The 1-dimensional case is particularly simple, but it still provides a valuable framework for understanding the basics of orthogonal structures. The orthogonal group, denoted as $O(V)$, consists of all linear transformations on $V$ that preserve the orthogonal form. In the 1-dimensional case, this group is relatively small, but its representations are still of interest in the context of Weil representations. The interaction between the symplectic space $W$ and the orthogonal space $V$ is where the magic of Weil representations truly begins. The Weil representation provides a bridge between these two seemingly different types of spaces, allowing us to transfer information and techniques from one to the other. This interplay is crucial in many areas of mathematics, including the study of automorphic forms and the representation theory of reductive groups. The dimensions of the spaces $W$ and $V$ play a significant role in the structure of the Weil representation. In our case, the simplicity of the 2-dimensional symplectic space and the 1-dimensional orthogonal space allows us to focus on the essential features of the representation without getting bogged down in technical complexities. However, the theory of Weil representations extends to higher-dimensional spaces, where the interplay between symplectic and orthogonal structures becomes even richer and more intricate.

Defining the Key Element: The Element 'e'

Central to our discussion is the element $e$. Let $e$ be an element in $F^{\times}/(F^{\times})^2$, where $F^{\times}$ denotes the multiplicative group of $F$, and $(F^{\times})^2$ represents the subgroup of squares. In essence, $e$ is a representative of a square class in $F$. Understanding this element is critical because it parametrizes different quadratic extensions of $F$, which in turn influence the structure of the Weil representation. The set $F^{\times}/(F^{\times})^2$ classifies the quadratic characters of $F^{\times}$, providing a way to distinguish between different quadratic forms and their associated orthogonal groups. Each element $e$ corresponds to a unique quadratic extension $F(\sqrt{e})$ of $F$, where $\sqrt{e}$ is a square root of $e$. These quadratic extensions play a crucial role in the arithmetic of $F$ and its extensions. The choice of $e$ affects the structure of the orthogonal space $V$. Since $V$ is 1-dimensional, its orthogonal form is determined by a single scalar, which can be chosen as $e$. This choice influences the orthogonal group $O(V)$ and its representations. The element $e$ also influences the way in which the symplectic space $W$ and the orthogonal space $V$ interact. The Weil representation is constructed by intertwining the representations of the symplectic group $Sp(W)$ and the orthogonal group $O(V)$, and the element $e$ plays a key role in this intertwining process. Specifically, the choice of $e$ determines the character of the Weil representation, which is a function that encodes important information about the representation. The character of a representation is a fundamental tool for studying its properties, such as its irreducibility and its decomposition into irreducible components. The character of the Weil representation is particularly sensitive to the choice of $e$, reflecting the intimate connection between the element $e$ and the structure of the representation. Moreover, the element $e$ appears in various formulas and identities related to the Weil representation, such as the Weil factor and the theta correspondence. These formulas highlight the central role of $e$ in the theory of Weil representations. The Weil factor, for example, is a complex number that depends on $e$ and the additive character of $F$, and it plays a crucial role in the construction of the Weil representation. The theta correspondence, on the other hand, is a deep connection between the representations of symplectic and orthogonal groups, and it is strongly influenced by the choice of $e$. In summary, the element $e$ is a key parameter that governs the structure and properties of the Weil representation. Its presence in the definition of the orthogonal space, the character of the representation, and various related formulas underscores its importance in the theory of Weil representations. Understanding the role of $e$ is essential for unraveling the intricacies of these representations and their applications in number theory and representation theory.

Metaplectic Group and its Significance

The metaplectic group, denoted as $\Mp$, is a central extension of the symplectic group $Sp(W)$. To truly appreciate its significance, one must delve into the nuances of group theory and representation theory. The symplectic group $Sp(W)$ consists of linear transformations that preserve a symplectic form on $W$. However, when constructing the Weil representation, we encounter certain ambiguities that necessitate a finer structure, leading us to the metaplectic group. The metaplectic group can be viewed as a “double cover” of the symplectic group. This means that there is a surjective homomorphism from $\Mp$ to $Sp(W)$ with a kernel of order 2. In other words, each element of $Sp(W)$ has two preimages in $\Mp$, reflecting an intrinsic ambiguity in the construction of the Weil representation. This ambiguity arises from the square roots that appear in the definition of the Weil representation. To resolve this, we lift the representation from $Sp(W)$ to its double cover, the metaplectic group $\Mp$. The central extension structure of $\Mp$ is crucial for understanding its representation theory. The kernel of the homomorphism from $\Mp$ to $Sp(W)$ is a central subgroup of order 2, typically denoted as $\{\pm 1\}$. This subgroup plays a significant role in the classification of the representations of $\Mp$. The representations of $\Mp$ are closely related to the representations of $Sp(W)$, but they exhibit certain unique features due to the central extension. In particular, the Weil representation is a representation of $\Mp$, not of $Sp(W)$, highlighting the necessity of working with the metaplectic group. The construction of the metaplectic group involves subtle topological and algebraic considerations. It can be defined using various approaches, such as cocycles or central extensions of topological groups. The precise construction depends on the field $F$ and the symplectic space $W$, but the underlying principle remains the same: to resolve the ambiguities in the definition of the Weil representation. The metaplectic group is not just an abstract algebraic object; it has profound connections to other areas of mathematics and physics. In number theory, it appears in the study of automorphic forms and the theta correspondence. In physics, it is related to the quantization of classical systems and the harmonic oscillator. The Weil representation of the metaplectic group provides a bridge between these different fields, allowing us to transfer techniques and insights from one to another. For instance, the theta correspondence is a deep connection between the representations of the metaplectic group and the representations of orthogonal groups. It arises from the study of theta functions, which are classical objects in number theory, and it has far-reaching consequences in representation theory. The theta correspondence can be viewed as a way to transfer representations between different groups, providing a powerful tool for understanding their structure. In summary, the metaplectic group is a fundamental object in the theory of Weil representations. Its central extension structure, its representations, and its connections to other areas of mathematics and physics make it a central player in the story of Weil representations. Understanding the metaplectic group is essential for grasping the intricacies of the Weil representation and its applications.

Delving into the Character of the Weil Representation

The character of a representation is a crucial tool for understanding its properties. It is a function that encodes essential information about the representation, such as its dimension, its irreducibility, and its decomposition into irreducible components. In the context of the Weil representation, the character plays a particularly important role, as it reflects the intricate interplay between the symplectic and orthogonal structures. Let $\chi$ denote the character of the Weil representation. It is a complex-valued function defined on the metaplectic group $\Mp$. For each element $g$ in $\Mp$, the value $\chi(g)$ is the trace of the operator corresponding to $g$ in the Weil representation. The character $\chi$ is a class function, meaning that it is constant on conjugacy classes of $\Mp$. This property simplifies the study of the character, as we only need to compute its values on representative elements of each conjugacy class. The character of the Weil representation is not just an abstract mathematical object; it has concrete formulas and explicit expressions. These formulas often involve special functions, such as Gauss sums and local factors, which reflect the arithmetic nature of the underlying field $F$. The explicit formulas for the character of the Weil representation are sensitive to the choice of the element $e$, which parametrizes the orthogonal space $V$. This sensitivity highlights the dependence of the Weil representation on the orthogonal structure. The character of the Weil representation can be used to determine its irreducibility. A representation is said to be irreducible if it cannot be decomposed into smaller, non-trivial subrepresentations. The character provides a criterion for irreducibility: a representation is irreducible if and only if the integral of the square of its character over the group is equal to 1. Applying this criterion to the Weil representation, we find that its irreducibility depends on the choice of the element $e$ and the characteristic of the field $F$. In some cases, the Weil representation is irreducible, while in others, it decomposes into a direct sum of irreducible components. The decomposition of the Weil representation is a central topic in the theory of automorphic forms. The irreducible components of the Weil representation often appear as building blocks in the construction of automorphic representations, which are fundamental objects in number theory. Understanding the character of the Weil representation is essential for unraveling the structure of these automorphic representations. The character of the Weil representation also plays a role in the theta correspondence. The theta correspondence is a deep connection between the representations of the metaplectic group and the representations of orthogonal groups. It arises from the study of theta functions, which are classical objects in number theory. The character of the Weil representation appears in the definition of the theta kernel, which is a function that intertwines the representations of the two groups. By analyzing the character of the Weil representation, we can gain insights into the theta correspondence and its applications. In summary, the character of the Weil representation is a powerful tool for studying its properties and its connections to other areas of mathematics. Its explicit formulas, its role in irreducibility criteria, and its appearance in the theta correspondence make it a central object in the theory of Weil representations. Understanding the character is essential for unraveling the intricacies of these representations and their applications in number theory and representation theory.

Character Dependence on 'e'

The character of the Weil representation exhibits a profound dependence on the element $e \in F^{\times}/(F^{\times})^2$. This dependence is not merely a technicality; it reflects the deep interplay between symplectic and orthogonal structures inherent in the Weil representation. The element $e$, as a representative of a square class in $F$, dictates the structure of the 1-dimensional orthogonal space $V$. Different choices of $e$ correspond to different orthogonal spaces, which in turn influence the way $V$ interacts with the 2-dimensional symplectic space $W$. This interaction is at the heart of the Weil representation, and its character captures this interplay in a concise and informative way. The character of the Weil representation can be expressed as a sum over certain cosets or subgroups, and the terms in this sum often involve quadratic characters or other arithmetic functions that depend explicitly on $e$. This explicit dependence makes the character a sensitive probe of the orthogonal structure. For example, the character may take different values depending on whether $e$ is a square or a non-square in $F$. This distinction reflects the fact that the orthogonal group $O(V)$ has different structures depending on the square class of $e$. When $e$ is a square, $O(V)$ is isomorphic to the multiplicative group $\{\pm 1\}$, while when $e$ is a non-square, $O(V)$ is a more complicated group. The character of the Weil representation can also be used to distinguish between different irreducible components of the representation. In some cases, the Weil representation decomposes into a direct sum of irreducible representations, and the character can help us identify these components. The decomposition pattern often depends on the element $e$, reflecting the fact that the Weil representation intertwines the representations of the symplectic group $Sp(W)$ and the orthogonal group $O(V)$. The character of the Weil representation is not only dependent on $e$, but also on other parameters, such as the additive character of $F$ and the element of the metaplectic group at which the character is evaluated. However, the dependence on $e$ is particularly significant because it highlights the role of the orthogonal structure in the Weil representation. The dependence of the character on $e$ has important consequences for the theta correspondence. The theta correspondence is a deep connection between the representations of the metaplectic group and the representations of orthogonal groups, and it arises from the study of theta functions. The character of the Weil representation appears in the definition of the theta kernel, which is a function that intertwines the representations of the two groups. The dependence of the character on $e$ implies that the theta correspondence will also depend on $e$. This dependence allows us to transfer information about the representations of orthogonal groups to the representations of the metaplectic group, and vice versa. In summary, the character of the Weil representation exhibits a profound dependence on the element $e$, reflecting the interplay between symplectic and orthogonal structures. This dependence has important consequences for the structure of the Weil representation, its irreducible components, and its role in the theta correspondence. Understanding this dependence is essential for unraveling the intricacies of the Weil representation and its applications in number theory and representation theory.

Conclusion

The character dependence of the Weil representation on the element $e$ unveils the intricate relationship between symplectic and orthogonal spaces in representation theory. This dependence is not a mere mathematical curiosity but a fundamental aspect that governs the structure and behavior of the Weil representation. The element $e$, representing a square class in the p-adic local field $F$, acts as a crucial parameter that shapes the orthogonal space $V$ and, consequently, influences the character of the Weil representation. The metaplectic group, a central extension of the symplectic group, plays a pivotal role in this context. Its representations, particularly the Weil representation, provide a bridge between symplectic and orthogonal groups, allowing us to study their interactions through the lens of representation theory. The character of the Weil representation, as a complex-valued function on the metaplectic group, serves as a fingerprint that encodes essential information about the representation. Its dependence on $e$ highlights the sensitivity of the Weil representation to the underlying orthogonal structure. This dependence manifests in explicit formulas for the character, which often involve arithmetic functions that depend on $e$. Furthermore, the character's dependence on $e$ has profound implications for the theta correspondence, a deep connection between the representations of the metaplectic group and orthogonal groups. The theta correspondence allows us to transfer information between these groups, and the dependence on $e$ ensures that this transfer respects the orthogonal structure. In conclusion, understanding the character dependence of the Weil representation on the element $e$ is paramount for researchers and enthusiasts in representation theory, number theory, and related fields. It provides a deeper appreciation for the intricate interplay between symplectic and orthogonal structures and opens avenues for further exploration and applications. The Weil representation, with its rich structure and far-reaching connections, continues to be a central object of study, offering insights into the fundamental symmetries and patterns that underlie the mathematical universe.