Here's a simple implementation of a GET /auth endpoint in FastAPI that authenticates a user:

```python
from fastapi import APIRouter, HTTPException

users = [
    {"username": "demo", "password": "password"}
]

router = APIRouter()

@router.get("/auth")
async def authenticate(
    username: str = "demo",
    password: str = "password"
):
    """Authenticate user"""
    if request.method != "GET":
        raise HTTPException(status_code=405, detail={
            "message": "Method Not Allowed",
            "method": request.method,
            "_verb": "get"
        })

    user = next((u for u in users if u["username"] == username), None)
    if not user or user["password"] != password:
        raise HTTPException(status_code=400, detail="Invalid credentials")

    return {
        "message": "Authentication successful",
        "method": "GET",
        "_verb": "get",
        "user": username,
        "token": "dummy_jwt_token_123"
    }
```

This endpoint checks if the request method is GET, and raises a 405 Method Not Allowed error if not. It then looks for a user with the provided username and password in the in-memory `users` list. If a matching user is found, it returns a successful authentication response with a dummy JWT token. Otherwise, it raises a 400 Bad Request error for invalid credentials.

The response includes the request method, a `_verb` field indicating the HTTP verb, the authenticated username, and a dummy JWT token.